Dirichlet series
| L(s) = 1 | − 4-s + 3·5-s − 5·9-s − 16·19-s − 3·20-s + 4·25-s − 12·29-s − 2·31-s + 5·36-s + 12·41-s − 15·45-s − 4·49-s − 18·59-s + 20·61-s − 4·64-s + 6·71-s + 16·76-s − 28·79-s + 16·81-s + 24·89-s − 48·95-s − 4·100-s + 24·101-s + 160·109-s + 12·116-s + 2·124-s + 15·125-s + ⋯ |
| L(s) = 1 | − 1/2·4-s + 1.34·5-s − 5/3·9-s − 3.67·19-s − 0.670·20-s + 4/5·25-s − 2.22·29-s − 0.359·31-s + 5/6·36-s + 1.87·41-s − 2.23·45-s − 4/7·49-s − 2.34·59-s + 2.56·61-s − 1/2·64-s + 0.712·71-s + 1.83·76-s − 3.15·79-s + 16/9·81-s + 2.54·89-s − 4.92·95-s − 2/5·100-s + 2.38·101-s + 15.3·109-s + 1.11·116-s + 0.179·124-s + 1.34·125-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{16} \cdot 11^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{16} \cdot 11^{32}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(5^{16} \cdot 11^{32}\) |
| Sign: | $1$ |
| Analytic conductor: | \(8.80075\times 10^{10}\) |
| Root analytic conductor: | \(2.19794\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((32,\ 5^{16} \cdot 11^{32} ,\ ( \ : [1/2]^{16} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(9.039616699\) |
| \(L(\frac12)\) | \(\approx\) | \(9.039616699\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 5 | \( 1 - 3 T + p T^{2} - 18 T^{3} + 54 T^{4} - 147 T^{5} + 74 p T^{6} - 882 T^{7} + 2021 T^{8} - 882 p T^{9} + 74 p^{3} T^{10} - 147 p^{3} T^{11} + 54 p^{4} T^{12} - 18 p^{5} T^{13} + p^{7} T^{14} - 3 p^{7} T^{15} + p^{8} T^{16} \) |
| 11 | \( 1 \) | |
| good | 2 | \( 1 + T^{2} + T^{4} + 5 T^{6} - 7 T^{8} + 5 p^{3} T^{10} + 11 p^{2} T^{12} - p^{6} T^{14} + 13 p^{4} T^{16} - p^{8} T^{18} + 11 p^{6} T^{20} + 5 p^{9} T^{22} - 7 p^{8} T^{24} + 5 p^{10} T^{26} + p^{12} T^{28} + p^{14} T^{30} + p^{16} T^{32} \) |
| 3 | \( ( 1 - T + p T^{2} - 8 T^{3} + 8 T^{4} + 7 T^{5} + 2 p T^{6} + 56 T^{7} - 137 T^{8} + 56 p T^{9} + 2 p^{3} T^{10} + 7 p^{3} T^{11} + 8 p^{4} T^{12} - 8 p^{5} T^{13} + p^{7} T^{14} - p^{7} T^{15} + p^{8} T^{16} )( 1 + T + p T^{2} + 8 T^{3} + 8 T^{4} - 7 T^{5} + 2 p T^{6} - 56 T^{7} - 137 T^{8} - 56 p T^{9} + 2 p^{3} T^{10} - 7 p^{3} T^{11} + 8 p^{4} T^{12} + 8 p^{5} T^{13} + p^{7} T^{14} + p^{7} T^{15} + p^{8} T^{16} ) \) | |
| 7 | \( ( 1 - 4 T + 9 T^{2} - 8 T^{3} - 31 T^{4} - 8 p T^{5} + 9 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2}( 1 + 4 T + 9 T^{2} + 8 T^{3} - 31 T^{4} + 8 p T^{5} + 9 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 13 | \( ( 1 + p T^{2} + p^{2} T^{4} + p^{3} T^{6} + p^{4} T^{8} )^{4} \) | |
| 17 | \( 1 + 40 T^{2} + 754 T^{4} + 7880 T^{6} + 56195 T^{8} - 743480 T^{10} - 49162204 T^{12} - 1346035360 T^{14} - 25538281211 T^{16} - 1346035360 p^{2} T^{18} - 49162204 p^{4} T^{20} - 743480 p^{6} T^{22} + 56195 p^{8} T^{24} + 7880 p^{10} T^{26} + 754 p^{12} T^{28} + 40 p^{14} T^{30} + p^{16} T^{32} \) | |
| 19 | \( ( 1 + 4 T - 3 T^{2} - 88 T^{3} - 295 T^{4} - 88 p T^{5} - 3 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{4} \) | |
| 23 | \( ( 1 - 85 T^{2} + 2856 T^{4} - 85 p^{2} T^{6} + p^{4} T^{8} )^{4} \) | |
| 29 | \( ( 1 + 6 T + 2 T^{2} - 18 T^{3} + 27 T^{4} + 13830 T^{5} + 77248 T^{6} + 13104 T^{7} - 164095 T^{8} + 13104 p T^{9} + 77248 p^{2} T^{10} + 13830 p^{3} T^{11} + 27 p^{4} T^{12} - 18 p^{5} T^{13} + 2 p^{6} T^{14} + 6 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 31 | \( ( 1 + T - 53 T^{2} - 76 T^{3} + 1856 T^{4} + 2065 T^{5} - 49582 T^{6} - 30520 T^{7} + 927307 T^{8} - 30520 p T^{9} - 49582 p^{2} T^{10} + 2065 p^{3} T^{11} + 1856 p^{4} T^{12} - 76 p^{5} T^{13} - 53 p^{6} T^{14} + p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 37 | \( ( 1 - 7 T + p T^{2} - 434 T^{3} + 3038 T^{4} - 14791 T^{5} + 3482 p T^{6} - 917042 T^{7} + 4545133 T^{8} - 917042 p T^{9} + 3482 p^{3} T^{10} - 14791 p^{3} T^{11} + 3038 p^{4} T^{12} - 434 p^{5} T^{13} + p^{7} T^{14} - 7 p^{7} T^{15} + p^{8} T^{16} )( 1 + 7 T + p T^{2} + 434 T^{3} + 3038 T^{4} + 14791 T^{5} + 3482 p T^{6} + 917042 T^{7} + 4545133 T^{8} + 917042 p T^{9} + 3482 p^{3} T^{10} + 14791 p^{3} T^{11} + 3038 p^{4} T^{12} + 434 p^{5} T^{13} + p^{7} T^{14} + 7 p^{7} T^{15} + p^{8} T^{16} ) \) | |
| 41 | \( ( 1 - 6 T - 22 T^{2} + 234 T^{3} - 333 T^{4} - 27510 T^{5} + 163792 T^{6} + 301392 T^{7} - 3981055 T^{8} + 301392 p T^{9} + 163792 p^{2} T^{10} - 27510 p^{3} T^{11} - 333 p^{4} T^{12} + 234 p^{5} T^{13} - 22 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 43 | \( ( 1 - 74 T^{2} + p^{2} T^{4} )^{8} \) | |
| 47 | \( ( 1 - 12 T + 97 T^{2} - 600 T^{3} + 2641 T^{4} - 600 p T^{5} + 97 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} )^{2}( 1 + 12 T + 97 T^{2} + 600 T^{3} + 2641 T^{4} + 600 p T^{5} + 97 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 53 | \( 1 + 100 T^{2} + 3994 T^{4} + 79700 T^{6} + 4181555 T^{8} + 601993300 T^{10} + 25958059556 T^{12} - 474038340400 T^{14} - 67202502041291 T^{16} - 474038340400 p^{2} T^{18} + 25958059556 p^{4} T^{20} + 601993300 p^{6} T^{22} + 4181555 p^{8} T^{24} + 79700 p^{10} T^{26} + 3994 p^{12} T^{28} + 100 p^{14} T^{30} + p^{16} T^{32} \) | |
| 59 | \( ( 1 + 9 T - 49 T^{2} - 1080 T^{3} - 2052 T^{4} + 5625 T^{5} - 85526 T^{6} + 1168884 T^{7} + 31768223 T^{8} + 1168884 p T^{9} - 85526 p^{2} T^{10} + 5625 p^{3} T^{11} - 2052 p^{4} T^{12} - 1080 p^{5} T^{13} - 49 p^{6} T^{14} + 9 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 61 | \( ( 1 - 10 T - 14 T^{2} + 670 T^{3} - 2725 T^{4} - 52330 T^{5} + 477344 T^{6} + 361360 T^{7} - 15969791 T^{8} + 361360 p T^{9} + 477344 p^{2} T^{10} - 52330 p^{3} T^{11} - 2725 p^{4} T^{12} + 670 p^{5} T^{13} - 14 p^{6} T^{14} - 10 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 67 | \( ( 1 - 181 T^{2} + 15312 T^{4} - 181 p^{2} T^{6} + p^{4} T^{8} )^{4} \) | |
| 71 | \( ( 1 - 3 T - 61 T^{2} + 180 T^{3} - 672 T^{4} - 58755 T^{5} + 492466 T^{6} + 1871208 T^{7} - 24183877 T^{8} + 1871208 p T^{9} + 492466 p^{2} T^{10} - 58755 p^{3} T^{11} - 672 p^{4} T^{12} + 180 p^{5} T^{13} - 61 p^{6} T^{14} - 3 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 73 | \( ( 1 + 98 T^{2} + 4275 T^{4} - 103292 T^{6} - 32904091 T^{8} - 103292 p^{2} T^{10} + 4275 p^{4} T^{12} + 98 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 79 | \( ( 1 + 14 T + 22 T^{2} - 1022 T^{3} - 8893 T^{4} + 35630 T^{5} + 778568 T^{6} + 1242976 T^{7} - 23161175 T^{8} + 1242976 p T^{9} + 778568 p^{2} T^{10} + 35630 p^{3} T^{11} - 8893 p^{4} T^{12} - 1022 p^{5} T^{13} + 22 p^{6} T^{14} + 14 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 83 | \( ( 1 + 122 T^{2} + 7995 T^{4} + 134932 T^{6} - 38615851 T^{8} + 134932 p^{2} T^{10} + 7995 p^{4} T^{12} + 122 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 89 | \( ( 1 - 3 T + 172 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{8} \) | |
| 97 | \( 1 + 337 T^{2} + 66433 T^{4} + 9673922 T^{6} + 1108767266 T^{8} + 124197193105 T^{10} + 14384725618562 T^{12} + 1652784116500610 T^{14} + 174600010053557857 T^{16} + 1652784116500610 p^{2} T^{18} + 14384725618562 p^{4} T^{20} + 124197193105 p^{6} T^{22} + 1108767266 p^{8} T^{24} + 9673922 p^{10} T^{26} + 66433 p^{12} T^{28} + 337 p^{14} T^{30} + p^{16} T^{32} \) | |
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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
−2.87666288599206460332784501004, −2.68147858476668304333879197572, −2.67222675711988413606763834955, −2.47205032111177118746078795717, −2.42220948014320926947089482305, −2.40303503296913315369193157958, −2.39382035894500244118004157514, −2.28289460668985161142602732134, −2.12093176153469214126668260302, −2.07706231510573367422423888403, −2.04846843308383035146166722836, −1.99646425150154740001297250200, −1.93292697102897325447017060690, −1.69491319047124218028586874241, −1.60445932126713260747907466429, −1.46653338037563866790197865700, −1.41622079260975717195869397326, −1.38787208085199242750838199426, −1.30645081002760085289416572556, −0.864458140676113926479147847971, −0.69855703504632046543322713791, −0.54086608481602934859280654824, −0.51410712858001655238797104350, −0.40865012325457688587998646827, −0.36361482955263125032393816576, 0.36361482955263125032393816576, 0.40865012325457688587998646827, 0.51410712858001655238797104350, 0.54086608481602934859280654824, 0.69855703504632046543322713791, 0.864458140676113926479147847971, 1.30645081002760085289416572556, 1.38787208085199242750838199426, 1.41622079260975717195869397326, 1.46653338037563866790197865700, 1.60445932126713260747907466429, 1.69491319047124218028586874241, 1.93292697102897325447017060690, 1.99646425150154740001297250200, 2.04846843308383035146166722836, 2.07706231510573367422423888403, 2.12093176153469214126668260302, 2.28289460668985161142602732134, 2.39382035894500244118004157514, 2.40303503296913315369193157958, 2.42220948014320926947089482305, 2.47205032111177118746078795717, 2.67222675711988413606763834955, 2.68147858476668304333879197572, 2.87666288599206460332784501004
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.