Dirichlet series
| L(s) = 1 | + 5·4-s + 5·9-s + 16·16-s − 18·19-s − 42·31-s + 25·36-s − 60·41-s + 4·49-s + 24·59-s + 30·61-s + 27·64-s − 90·76-s + 58·79-s − 8·81-s + 6·89-s − 6·101-s + 40·109-s − 32·121-s − 210·124-s + 127-s + 131-s + 137-s + 139-s + 80·144-s + 149-s + 151-s + 157-s + ⋯ |
| L(s) = 1 | + 5/2·4-s + 5/3·9-s + 4·16-s − 4.12·19-s − 7.54·31-s + 25/6·36-s − 9.37·41-s + 4/7·49-s + 3.12·59-s + 3.84·61-s + 27/8·64-s − 10.3·76-s + 6.52·79-s − 8/9·81-s + 0.635·89-s − 0.597·101-s + 3.83·109-s − 2.90·121-s − 18.8·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 20/3·144-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 5^{32} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 5^{32} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(3^{16} \cdot 5^{32} \cdot 7^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(9.09876\times 10^{9}\) |
| Root analytic conductor: | \(2.04747\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((32,\ 3^{16} \cdot 5^{32} \cdot 7^{16} ,\ ( \ : [1/2]^{16} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(2.177035398\) |
| \(L(\frac12)\) | \(\approx\) | \(2.177035398\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( 1 - 5 T^{2} + 11 p T^{4} - 110 T^{6} + 430 T^{8} - 110 p^{2} T^{10} + 11 p^{5} T^{12} - 5 p^{6} T^{14} + p^{8} T^{16} \) |
| 5 | \( 1 \) | |
| 7 | \( 1 - 4 T^{2} - 26 T^{4} - 244 T^{6} + 3907 T^{8} - 244 p^{2} T^{10} - 26 p^{4} T^{12} - 4 p^{6} T^{14} + p^{8} T^{16} \) | |
| good | 2 | \( 1 - 5 T^{2} + 9 T^{4} + p^{3} T^{6} - p^{6} T^{8} + 15 p^{3} T^{10} - p^{5} T^{12} - 23 p^{4} T^{14} + 69 p^{4} T^{16} - 23 p^{6} T^{18} - p^{9} T^{20} + 15 p^{9} T^{22} - p^{14} T^{24} + p^{13} T^{26} + 9 p^{12} T^{28} - 5 p^{14} T^{30} + p^{16} T^{32} \) |
| 11 | \( ( 1 + 16 T^{2} - 2 T^{4} + 30 T^{5} + 268 T^{6} + 1548 T^{7} + 21079 T^{8} + 1548 p T^{9} + 268 p^{2} T^{10} + 30 p^{3} T^{11} - 2 p^{4} T^{12} + 16 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 13 | \( ( 1 + 83 T^{2} + 3217 T^{4} + 76058 T^{6} + 1197778 T^{8} + 76058 p^{2} T^{10} + 3217 p^{4} T^{12} + 83 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 17 | \( 1 + 76 T^{2} + 3024 T^{4} + 4384 p T^{6} + 1179962 T^{8} + 10465632 T^{10} - 4424000 T^{12} - 1753904276 T^{14} - 37267323453 T^{16} - 1753904276 p^{2} T^{18} - 4424000 p^{4} T^{20} + 10465632 p^{6} T^{22} + 1179962 p^{8} T^{24} + 4384 p^{11} T^{26} + 3024 p^{12} T^{28} + 76 p^{14} T^{30} + p^{16} T^{32} \) | |
| 19 | \( ( 1 + 9 T + 100 T^{2} + 657 T^{3} + 4723 T^{4} + 26244 T^{5} + 148996 T^{6} + 704196 T^{7} + 3331528 T^{8} + 704196 p T^{9} + 148996 p^{2} T^{10} + 26244 p^{3} T^{11} + 4723 p^{4} T^{12} + 657 p^{5} T^{13} + 100 p^{6} T^{14} + 9 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 23 | \( 1 - 53 T^{2} + 399 T^{4} + 796 p T^{6} - 55297 T^{8} - 3303501 T^{10} - 170695622 T^{12} - 2962951637 T^{14} + 281105182152 T^{16} - 2962951637 p^{2} T^{18} - 170695622 p^{4} T^{20} - 3303501 p^{6} T^{22} - 55297 p^{8} T^{24} + 796 p^{11} T^{26} + 399 p^{12} T^{28} - 53 p^{14} T^{30} + p^{16} T^{32} \) | |
| 29 | \( ( 1 - 53 T^{2} + 3250 T^{4} - 128951 T^{6} + 4063174 T^{8} - 128951 p^{2} T^{10} + 3250 p^{4} T^{12} - 53 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 31 | \( ( 1 + 21 T + 262 T^{2} + 2415 T^{3} + 17293 T^{4} + 101304 T^{5} + 505090 T^{6} + 2328618 T^{7} + 11769748 T^{8} + 2328618 p T^{9} + 505090 p^{2} T^{10} + 101304 p^{3} T^{11} + 17293 p^{4} T^{12} + 2415 p^{5} T^{13} + 262 p^{6} T^{14} + 21 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 37 | \( 1 + 97 T^{2} + 5532 T^{4} + 186305 T^{6} + 3327797 T^{8} + 12794040 T^{10} + 1243533586 T^{12} + 183679377646 T^{14} + 10522943320200 T^{16} + 183679377646 p^{2} T^{18} + 1243533586 p^{4} T^{20} + 12794040 p^{6} T^{22} + 3327797 p^{8} T^{24} + 186305 p^{10} T^{26} + 5532 p^{12} T^{28} + 97 p^{14} T^{30} + p^{16} T^{32} \) | |
| 41 | \( ( 1 + 15 T + 218 T^{2} + 1791 T^{3} + 14136 T^{4} + 1791 p T^{5} + 218 p^{2} T^{6} + 15 p^{3} T^{7} + p^{4} T^{8} )^{4} \) | |
| 43 | \( ( 1 - 304 T^{2} + 41758 T^{4} - 3395548 T^{6} + 179098699 T^{8} - 3395548 p^{2} T^{10} + 41758 p^{4} T^{12} - 304 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 47 | \( 1 + 268 T^{2} + 36732 T^{4} + 3628040 T^{6} + 294880202 T^{8} + 20635855860 T^{10} + 1272375350416 T^{12} + 70342526859604 T^{14} + 3494142383383395 T^{16} + 70342526859604 p^{2} T^{18} + 1272375350416 p^{4} T^{20} + 20635855860 p^{6} T^{22} + 294880202 p^{8} T^{24} + 3628040 p^{10} T^{26} + 36732 p^{12} T^{28} + 268 p^{14} T^{30} + p^{16} T^{32} \) | |
| 53 | \( 1 - 104 T^{2} + 3204 T^{4} + 8144 T^{6} - 2651158 T^{8} - 214376472 T^{10} + 18555084688 T^{12} + 949831131304 T^{14} - 128029390162317 T^{16} + 949831131304 p^{2} T^{18} + 18555084688 p^{4} T^{20} - 214376472 p^{6} T^{22} - 2651158 p^{8} T^{24} + 8144 p^{10} T^{26} + 3204 p^{12} T^{28} - 104 p^{14} T^{30} + p^{16} T^{32} \) | |
| 59 | \( ( 1 - 12 T - 80 T^{2} + 1164 T^{3} + 7690 T^{4} - 80082 T^{5} - 434420 T^{6} + 1772232 T^{7} + 28861927 T^{8} + 1772232 p T^{9} - 434420 p^{2} T^{10} - 80082 p^{3} T^{11} + 7690 p^{4} T^{12} + 1164 p^{5} T^{13} - 80 p^{6} T^{14} - 12 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 61 | \( ( 1 - 15 T + 223 T^{2} - 2220 T^{3} + 19711 T^{4} - 141723 T^{5} + 816310 T^{6} - 5175267 T^{7} + 31433836 T^{8} - 5175267 p T^{9} + 816310 p^{2} T^{10} - 141723 p^{3} T^{11} + 19711 p^{4} T^{12} - 2220 p^{5} T^{13} + 223 p^{6} T^{14} - 15 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 67 | \( 1 + 484 T^{2} + 128562 T^{4} + 23999600 T^{6} + 3478000361 T^{8} + 410568162660 T^{10} + 40599889512562 T^{12} + 3416547531270040 T^{14} + 246775367829948324 T^{16} + 3416547531270040 p^{2} T^{18} + 40599889512562 p^{4} T^{20} + 410568162660 p^{6} T^{22} + 3478000361 p^{8} T^{24} + 23999600 p^{10} T^{26} + 128562 p^{12} T^{28} + 484 p^{14} T^{30} + p^{16} T^{32} \) | |
| 71 | \( ( 1 - 464 T^{2} + 99532 T^{4} - 12936548 T^{6} + 1114829374 T^{8} - 12936548 p^{2} T^{10} + 99532 p^{4} T^{12} - 464 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 73 | \( 1 - 335 T^{2} + 63708 T^{4} - 7606879 T^{6} + 587709533 T^{8} - 270827976 p T^{10} - 1697574900854 T^{12} + 348751235496070 T^{14} - 32202831466789560 T^{16} + 348751235496070 p^{2} T^{18} - 1697574900854 p^{4} T^{20} - 270827976 p^{7} T^{22} + 587709533 p^{8} T^{24} - 7606879 p^{10} T^{26} + 63708 p^{12} T^{28} - 335 p^{14} T^{30} + p^{16} T^{32} \) | |
| 79 | \( ( 1 - 29 T + 294 T^{2} - 25 p T^{3} + 27377 T^{4} - 260496 T^{5} + 598654 T^{6} - 2403434 T^{7} + 77714340 T^{8} - 2403434 p T^{9} + 598654 p^{2} T^{10} - 260496 p^{3} T^{11} + 27377 p^{4} T^{12} - 25 p^{6} T^{13} + 294 p^{6} T^{14} - 29 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 83 | \( ( 1 - 535 T^{2} + 130150 T^{4} - 19183249 T^{6} + 1908109846 T^{8} - 19183249 p^{2} T^{10} + 130150 p^{4} T^{12} - 535 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 89 | \( ( 1 - 3 T - 53 T^{2} - 2820 T^{3} + 14227 T^{4} + 160275 T^{5} + 3467116 T^{6} - 26593569 T^{7} - 193500020 T^{8} - 26593569 p T^{9} + 3467116 p^{2} T^{10} + 160275 p^{3} T^{11} + 14227 p^{4} T^{12} - 2820 p^{5} T^{13} - 53 p^{6} T^{14} - 3 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 97 | \( ( 1 + 368 T^{2} + 81676 T^{4} + 12257504 T^{6} + 1385094598 T^{8} + 12257504 p^{2} T^{10} + 81676 p^{4} T^{12} + 368 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
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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
−2.83660945168169740363129769845, −2.75816509400246229632862006474, −2.75012672877664786032607930903, −2.59821955061013722529315782727, −2.59055055806221664196223556217, −2.44104008240702619781872913263, −2.35582075671386229767245906980, −2.21814326971787094854441094236, −2.18796360531377195607005433481, −2.12329146800151951689482037386, −2.06593799670428155966886518039, −1.98906142220113348199396362561, −1.81237004585431757951152659651, −1.71491983708777816774014428603, −1.70563378181283139576660452266, −1.68626428707863088681843289657, −1.68127240916617916848456850759, −1.46039126370814782193244410126, −1.26591152543498665868978589482, −1.14860385572516273111740406141, −1.02789949496864581386411826587, −0.913089128559564419405041608863, −0.34990047450075523339949242913, −0.32915143197285837171937159843, −0.16304648905257452743268788615, 0.16304648905257452743268788615, 0.32915143197285837171937159843, 0.34990047450075523339949242913, 0.913089128559564419405041608863, 1.02789949496864581386411826587, 1.14860385572516273111740406141, 1.26591152543498665868978589482, 1.46039126370814782193244410126, 1.68127240916617916848456850759, 1.68626428707863088681843289657, 1.70563378181283139576660452266, 1.71491983708777816774014428603, 1.81237004585431757951152659651, 1.98906142220113348199396362561, 2.06593799670428155966886518039, 2.12329146800151951689482037386, 2.18796360531377195607005433481, 2.21814326971787094854441094236, 2.35582075671386229767245906980, 2.44104008240702619781872913263, 2.59055055806221664196223556217, 2.59821955061013722529315782727, 2.75012672877664786032607930903, 2.75816509400246229632862006474, 2.83660945168169740363129769845
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.