Properties

Degree 2
Conductor $ 1 $
Sign $-1$
Motivic weight 73
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.59e11·2-s + 2.55e16·3-s + 1.59e22·4-s + 2.70e25·5-s − 4.06e27·6-s − 5.83e30·7-s − 1.04e33·8-s − 6.69e34·9-s − 4.31e36·10-s + 1.33e38·11-s + 4.07e38·12-s − 6.53e40·13-s + 9.30e41·14-s + 6.91e41·15-s + 1.51e43·16-s + 1.34e45·17-s + 1.06e46·18-s + 4.71e46·19-s + 4.32e47·20-s − 1.48e47·21-s − 2.12e49·22-s − 7.28e48·23-s − 2.65e49·24-s − 3.25e50·25-s + 1.04e52·26-s − 3.43e51·27-s − 9.32e52·28-s + ⋯
L(s)  = 1  − 1.64·2-s + 0.0981·3-s + 1.69·4-s + 0.832·5-s − 0.161·6-s − 0.832·7-s − 1.13·8-s − 0.990·9-s − 1.36·10-s + 1.29·11-s + 0.166·12-s − 1.43·13-s + 1.36·14-s + 0.0817·15-s + 0.169·16-s + 1.65·17-s + 1.62·18-s + 0.997·19-s + 1.40·20-s − 0.0816·21-s − 2.13·22-s − 0.144·23-s − 0.111·24-s − 0.307·25-s + 2.35·26-s − 0.195·27-s − 1.40·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & -\,\Lambda(74-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+73/2) \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(73\)
character  :  $\chi_{1} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(1\)
Selberg data  =  \((2,\ 1,\ (\ :73/2),\ -1)\)
\(L(37)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{75}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where,\(F_p(T)\) is a polynomial of degree 2.
$p$$F_p(T)$
good2 \( 1 + 1.59e11T + 9.44e21T^{2} \)
3 \( 1 - 2.55e16T + 6.75e34T^{2} \)
5 \( 1 - 2.70e25T + 1.05e51T^{2} \)
7 \( 1 + 5.83e30T + 4.92e61T^{2} \)
11 \( 1 - 1.33e38T + 1.05e76T^{2} \)
13 \( 1 + 6.53e40T + 2.07e81T^{2} \)
17 \( 1 - 1.34e45T + 6.64e89T^{2} \)
19 \( 1 - 4.71e46T + 2.23e93T^{2} \)
23 \( 1 + 7.28e48T + 2.54e99T^{2} \)
29 \( 1 + 1.30e53T + 5.68e106T^{2} \)
31 \( 1 + 6.62e53T + 7.40e108T^{2} \)
37 \( 1 - 1.59e57T + 3.01e114T^{2} \)
41 \( 1 + 1.09e59T + 5.41e117T^{2} \)
43 \( 1 - 6.52e59T + 1.75e119T^{2} \)
47 \( 1 - 6.06e60T + 1.15e122T^{2} \)
53 \( 1 + 1.02e63T + 7.44e125T^{2} \)
59 \( 1 + 2.47e64T + 1.87e129T^{2} \)
61 \( 1 + 8.75e64T + 2.13e130T^{2} \)
67 \( 1 + 2.58e66T + 2.01e133T^{2} \)
71 \( 1 + 4.91e67T + 1.38e135T^{2} \)
73 \( 1 + 1.02e68T + 1.05e136T^{2} \)
79 \( 1 + 2.22e69T + 3.36e138T^{2} \)
83 \( 1 + 1.44e70T + 1.23e140T^{2} \)
89 \( 1 - 3.70e70T + 2.02e142T^{2} \)
97 \( 1 - 3.38e71T + 1.08e145T^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−16.63090389061215306283591218090, −14.30870191502070443638994056210, −11.87254265856493444085726676735, −9.892859099197717383933192707391, −9.268826832423104002057256829430, −7.47979629862534823040583339353, −5.92356223185394185889005904993, −2.90788868250565303219938558231, −1.41489081260714454570483608629, 0, 1.41489081260714454570483608629, 2.90788868250565303219938558231, 5.92356223185394185889005904993, 7.47979629862534823040583339353, 9.268826832423104002057256829430, 9.892859099197717383933192707391, 11.87254265856493444085726676735, 14.30870191502070443638994056210, 16.63090389061215306283591218090

Graph of the $Z$-function along the critical line