Dirichlet series
| L(s) = 1 | + 3·9-s − 32·25-s − 12·29-s − 2·37-s + 4·43-s − 36·53-s + 14·67-s + 20·79-s + 18·81-s + 6·107-s + 10·109-s + 90·113-s + 86·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 56·169-s + 173-s + 179-s + 181-s + 191-s + ⋯ |
| L(s) = 1 | + 9-s − 6.39·25-s − 2.22·29-s − 0.328·37-s + 0.609·43-s − 4.94·53-s + 1.71·67-s + 2.25·79-s + 2·81-s + 0.580·107-s + 0.957·109-s + 8.46·113-s + 7.81·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 4.30·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{32} \cdot 7^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{32} \cdot 7^{32}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(2^{32} \cdot 3^{32} \cdot 7^{32}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.40110\times 10^{18}\) |
| Root analytic conductor: | \(3.75308\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((32,\ 2^{32} \cdot 3^{32} \cdot 7^{32} ,\ ( \ : [1/2]^{16} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(5.447929762\) |
| \(L(\frac12)\) | \(\approx\) | \(5.447929762\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - p T^{2} - p^{2} T^{4} - p^{2} T^{6} + 25 p^{2} T^{8} - p^{4} T^{10} - p^{6} T^{12} - p^{7} T^{14} + p^{8} T^{16} \) | |
| 7 | \( 1 \) | |
| good | 5 | \( ( 1 + 16 T^{2} + 97 T^{4} + 277 T^{6} + 679 T^{8} + 277 p^{2} T^{10} + 97 p^{4} T^{12} + 16 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 11 | \( ( 1 - 43 T^{2} + 1057 T^{4} - 17899 T^{6} + 227137 T^{8} - 17899 p^{2} T^{10} + 1057 p^{4} T^{12} - 43 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 13 | \( 1 + 56 T^{2} + 1590 T^{4} + 26848 T^{6} + 274865 T^{8} + 1615656 T^{10} + 12863830 T^{12} + 363405056 T^{14} + 6601006116 T^{16} + 363405056 p^{2} T^{18} + 12863830 p^{4} T^{20} + 1615656 p^{6} T^{22} + 274865 p^{8} T^{24} + 26848 p^{10} T^{26} + 1590 p^{12} T^{28} + 56 p^{14} T^{30} + p^{16} T^{32} \) | |
| 17 | \( 1 - 58 T^{2} + 1761 T^{4} - 32948 T^{6} + 331181 T^{8} + 945588 T^{10} - 132300833 T^{12} + 3706114724 T^{14} - 71840565063 T^{16} + 3706114724 p^{2} T^{18} - 132300833 p^{4} T^{20} + 945588 p^{6} T^{22} + 331181 p^{8} T^{24} - 32948 p^{10} T^{26} + 1761 p^{12} T^{28} - 58 p^{14} T^{30} + p^{16} T^{32} \) | |
| 19 | \( 1 + 77 T^{2} + 2895 T^{4} + 64882 T^{6} + 896840 T^{8} + 7384440 T^{10} + 66022636 T^{12} + 2128839191 T^{14} + 54949527117 T^{16} + 2128839191 p^{2} T^{18} + 66022636 p^{4} T^{20} + 7384440 p^{6} T^{22} + 896840 p^{8} T^{24} + 64882 p^{10} T^{26} + 2895 p^{12} T^{28} + 77 p^{14} T^{30} + p^{16} T^{32} \) | |
| 23 | \( ( 1 - 103 T^{2} + 5785 T^{4} - 216259 T^{6} + 5824729 T^{8} - 216259 p^{2} T^{10} + 5785 p^{4} T^{12} - 103 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 29 | \( ( 1 + 6 T + 53 T^{2} + 246 T^{3} + 20 p T^{4} + 540 T^{5} - 22066 T^{6} - 299625 T^{7} - 1268282 T^{8} - 299625 p T^{9} - 22066 p^{2} T^{10} + 540 p^{3} T^{11} + 20 p^{5} T^{12} + 246 p^{5} T^{13} + 53 p^{6} T^{14} + 6 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 31 | \( 1 + 176 T^{2} + 16407 T^{4} + 1045570 T^{6} + 50906954 T^{8} + 2036295270 T^{10} + 71233973380 T^{12} + 2306080860179 T^{14} + 72156636517806 T^{16} + 2306080860179 p^{2} T^{18} + 71233973380 p^{4} T^{20} + 2036295270 p^{6} T^{22} + 50906954 p^{8} T^{24} + 1045570 p^{10} T^{26} + 16407 p^{12} T^{28} + 176 p^{14} T^{30} + p^{16} T^{32} \) | |
| 37 | \( ( 1 + T - 81 T^{2} - 94 T^{3} + 2696 T^{4} + 2592 T^{5} - 94076 T^{6} - 24353 T^{7} + 4259061 T^{8} - 24353 p T^{9} - 94076 p^{2} T^{10} + 2592 p^{3} T^{11} + 2696 p^{4} T^{12} - 94 p^{5} T^{13} - 81 p^{6} T^{14} + p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 41 | \( 1 - 151 T^{2} + 9564 T^{4} - 353897 T^{6} + 12352577 T^{8} - 588962616 T^{10} + 30503470855 T^{12} - 1573469011633 T^{14} + 72313421958936 T^{16} - 1573469011633 p^{2} T^{18} + 30503470855 p^{4} T^{20} - 588962616 p^{6} T^{22} + 12352577 p^{8} T^{24} - 353897 p^{10} T^{26} + 9564 p^{12} T^{28} - 151 p^{14} T^{30} + p^{16} T^{32} \) | |
| 43 | \( ( 1 - 2 T - 93 T^{2} - 64 T^{3} + 92 p T^{4} + 12192 T^{5} - 158990 T^{6} - 321485 T^{7} + 7972668 T^{8} - 321485 p T^{9} - 158990 p^{2} T^{10} + 12192 p^{3} T^{11} + 92 p^{5} T^{12} - 64 p^{5} T^{13} - 93 p^{6} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 47 | \( 1 - 154 T^{2} + 7683 T^{4} - 245888 T^{6} + 25028264 T^{8} - 1538143278 T^{10} + 38475997042 T^{12} - 55371690107 p T^{14} + 205422570296046 T^{16} - 55371690107 p^{3} T^{18} + 38475997042 p^{4} T^{20} - 1538143278 p^{6} T^{22} + 25028264 p^{8} T^{24} - 245888 p^{10} T^{26} + 7683 p^{12} T^{28} - 154 p^{14} T^{30} + p^{16} T^{32} \) | |
| 53 | \( ( 1 + 18 T + 167 T^{2} + 1062 T^{3} + 3655 T^{4} - 5670 T^{5} - 78073 T^{6} - 140364 T^{7} + 103069 T^{8} - 140364 p T^{9} - 78073 p^{2} T^{10} - 5670 p^{3} T^{11} + 3655 p^{4} T^{12} + 1062 p^{5} T^{13} + 167 p^{6} T^{14} + 18 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 59 | \( 1 - 376 T^{2} + 75453 T^{4} - 10695122 T^{6} + 1193881406 T^{8} - 111003832932 T^{10} + 8882998756942 T^{12} - 624280670070511 T^{14} + 38988958528434078 T^{16} - 624280670070511 p^{2} T^{18} + 8882998756942 p^{4} T^{20} - 111003832932 p^{6} T^{22} + 1193881406 p^{8} T^{24} - 10695122 p^{10} T^{26} + 75453 p^{12} T^{28} - 376 p^{14} T^{30} + p^{16} T^{32} \) | |
| 61 | \( 1 + 137 T^{2} + 306 T^{4} - 368483 T^{6} + 33183791 T^{8} + 2225199582 T^{10} - 143899132121 T^{12} + 1205057298539 T^{14} + 1133086682080440 T^{16} + 1205057298539 p^{2} T^{18} - 143899132121 p^{4} T^{20} + 2225199582 p^{6} T^{22} + 33183791 p^{8} T^{24} - 368483 p^{10} T^{26} + 306 p^{12} T^{28} + 137 p^{14} T^{30} + p^{16} T^{32} \) | |
| 67 | \( ( 1 - 7 T - 108 T^{2} - 293 T^{3} + 11753 T^{4} + 50778 T^{5} - 198827 T^{6} - 2620663 T^{7} - 10701486 T^{8} - 2620663 p T^{9} - 198827 p^{2} T^{10} + 50778 p^{3} T^{11} + 11753 p^{4} T^{12} - 293 p^{5} T^{13} - 108 p^{6} T^{14} - 7 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 71 | \( ( 1 - 361 T^{2} + 64216 T^{4} - 7488268 T^{6} + 623512600 T^{8} - 7488268 p^{2} T^{10} + 64216 p^{4} T^{12} - 361 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 73 | \( 1 + 341 T^{2} + 53883 T^{4} + 6153142 T^{6} + 667124744 T^{8} + 70476881112 T^{10} + 6578299196452 T^{12} + 524328155054171 T^{14} + 38507998242556701 T^{16} + 524328155054171 p^{2} T^{18} + 6578299196452 p^{4} T^{20} + 70476881112 p^{6} T^{22} + 667124744 p^{8} T^{24} + 6153142 p^{10} T^{26} + 53883 p^{12} T^{28} + 341 p^{14} T^{30} + p^{16} T^{32} \) | |
| 79 | \( ( 1 - 10 T - 123 T^{2} + 844 T^{3} + 11042 T^{4} + 2166 T^{5} - 1268066 T^{6} - 531391 T^{7} + 106354368 T^{8} - 531391 p T^{9} - 1268066 p^{2} T^{10} + 2166 p^{3} T^{11} + 11042 p^{4} T^{12} + 844 p^{5} T^{13} - 123 p^{6} T^{14} - 10 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 83 | \( 1 - 397 T^{2} + 87990 T^{4} - 12501089 T^{6} + 1185510017 T^{8} - 57837981066 T^{10} - 2507933243939 T^{12} + 849901181197043 T^{14} - 92913215010718536 T^{16} + 849901181197043 p^{2} T^{18} - 2507933243939 p^{4} T^{20} - 57837981066 p^{6} T^{22} + 1185510017 p^{8} T^{24} - 12501089 p^{10} T^{26} + 87990 p^{12} T^{28} - 397 p^{14} T^{30} + p^{16} T^{32} \) | |
| 89 | \( 1 - 64 T^{2} - 5391 T^{4} + 717502 T^{6} - 95745073 T^{8} + 6819998976 T^{10} - 66652775855 T^{12} - 65309085806572 T^{14} + 11564582627289681 T^{16} - 65309085806572 p^{2} T^{18} - 66652775855 p^{4} T^{20} + 6819998976 p^{6} T^{22} - 95745073 p^{8} T^{24} + 717502 p^{10} T^{26} - 5391 p^{12} T^{28} - 64 p^{14} T^{30} + p^{16} T^{32} \) | |
| 97 | \( 1 + 404 T^{2} + 81609 T^{4} + 10963666 T^{6} + 1046175986 T^{8} + 591335472 p T^{10} - 1663618077194 T^{12} - 785812116013063 T^{14} - 99406691908304130 T^{16} - 785812116013063 p^{2} T^{18} - 1663618077194 p^{4} T^{20} + 591335472 p^{7} T^{22} + 1046175986 p^{8} T^{24} + 10963666 p^{10} T^{26} + 81609 p^{12} T^{28} + 404 p^{14} T^{30} + p^{16} T^{32} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.22927921605963506852372017890, −2.18597100066109821771850863662, −2.16362870499009616234400164315, −2.11495508166530408878188395916, −2.08456354377139491549679178558, −1.98525447101084030078269161331, −1.86870171444045531589921975800, −1.80263247179879798009296174856, −1.77341486006727847717514540371, −1.65547490398817297986993509959, −1.61318186668239508539772981759, −1.60067901859861158981395337608, −1.46833835600118240054427855142, −1.31272473568827118394717971471, −1.25533628024403701933132743221, −1.20793033416186110115444999176, −1.08899242454003978017365315579, −0.886313233136152138190020095856, −0.67773504726193214925363301324, −0.62981409522322194286450247315, −0.59032168973052014859751601801, −0.52398223206591902019804255344, −0.37539087278389075182465950333, −0.20815553163180755221691102674, −0.14325553293528411795140695628, 0.14325553293528411795140695628, 0.20815553163180755221691102674, 0.37539087278389075182465950333, 0.52398223206591902019804255344, 0.59032168973052014859751601801, 0.62981409522322194286450247315, 0.67773504726193214925363301324, 0.886313233136152138190020095856, 1.08899242454003978017365315579, 1.20793033416186110115444999176, 1.25533628024403701933132743221, 1.31272473568827118394717971471, 1.46833835600118240054427855142, 1.60067901859861158981395337608, 1.61318186668239508539772981759, 1.65547490398817297986993509959, 1.77341486006727847717514540371, 1.80263247179879798009296174856, 1.86870171444045531589921975800, 1.98525447101084030078269161331, 2.08456354377139491549679178558, 2.11495508166530408878188395916, 2.16362870499009616234400164315, 2.18597100066109821771850863662, 2.22927921605963506852372017890
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.