Properties

Degree $2$
Conductor $1$
Sign $-1$
Motivic weight $37$
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4.80e5·2-s + 3.45e7·3-s + 9.33e10·4-s + 1.38e13·5-s − 1.66e13·6-s − 5.42e15·7-s + 2.11e16·8-s − 4.49e17·9-s − 6.64e18·10-s − 1.03e18·11-s + 3.22e18·12-s − 1.23e19·13-s + 2.60e21·14-s + 4.78e20·15-s − 2.30e22·16-s − 5.41e22·17-s + 2.15e23·18-s − 3.08e23·19-s + 1.29e24·20-s − 1.87e23·21-s + 4.96e23·22-s − 2.61e25·23-s + 7.32e23·24-s + 1.18e26·25-s + 5.92e24·26-s − 3.11e25·27-s − 5.06e26·28-s + ⋯
L(s)  = 1  − 1.29·2-s + 0.0515·3-s + 0.679·4-s + 1.62·5-s − 0.0667·6-s − 1.25·7-s + 0.415·8-s − 0.997·9-s − 2.10·10-s − 0.0560·11-s + 0.0350·12-s − 0.0303·13-s + 1.63·14-s + 0.0835·15-s − 1.21·16-s − 0.934·17-s + 1.29·18-s − 0.680·19-s + 1.10·20-s − 0.0648·21-s + 0.0725·22-s − 1.68·23-s + 0.0214·24-s + 1.62·25-s + 0.0393·26-s − 0.102·27-s − 0.854·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & -\,\Lambda(38-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+37/2) \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1\)
Sign: $-1$
Motivic weight: \(37\)
Character: $\chi_{1} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1,\ (\ :37/2),\ -1)\)

Particular Values

\(L(19)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{39}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
good2 \( 1 + 4.80e5T + 1.37e11T^{2} \)
3 \( 1 - 3.45e7T + 4.50e17T^{2} \)
5 \( 1 - 1.38e13T + 7.27e25T^{2} \)
7 \( 1 + 5.42e15T + 1.85e31T^{2} \)
11 \( 1 + 1.03e18T + 3.40e38T^{2} \)
13 \( 1 + 1.23e19T + 1.64e41T^{2} \)
17 \( 1 + 5.41e22T + 3.36e45T^{2} \)
19 \( 1 + 3.08e23T + 2.06e47T^{2} \)
23 \( 1 + 2.61e25T + 2.42e50T^{2} \)
29 \( 1 - 2.49e26T + 1.28e54T^{2} \)
31 \( 1 + 2.34e27T + 1.51e55T^{2} \)
37 \( 1 - 6.21e28T + 1.05e58T^{2} \)
41 \( 1 + 8.53e29T + 4.70e59T^{2} \)
43 \( 1 - 3.53e29T + 2.74e60T^{2} \)
47 \( 1 - 6.68e29T + 7.37e61T^{2} \)
53 \( 1 - 1.24e32T + 6.28e63T^{2} \)
59 \( 1 - 6.57e31T + 3.32e65T^{2} \)
61 \( 1 + 1.06e33T + 1.14e66T^{2} \)
67 \( 1 + 8.44e33T + 3.67e67T^{2} \)
71 \( 1 + 7.04e33T + 3.13e68T^{2} \)
73 \( 1 - 9.36e33T + 8.76e68T^{2} \)
79 \( 1 - 1.51e35T + 1.63e70T^{2} \)
83 \( 1 + 1.43e35T + 1.01e71T^{2} \)
89 \( 1 - 2.28e35T + 1.34e72T^{2} \)
97 \( 1 - 7.07e36T + 3.24e73T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−22.23571281374329264203523647512, −19.87494481669384388760742054607, −18.00274185120459235642841690396, −16.71303239842308479600166267384, −13.54710833338065349296574072216, −10.20458218628807267645251886245, −8.972504432680970898032209122777, −6.23021964751350246881882725271, −2.19691573656503192005496061157, 0, 2.19691573656503192005496061157, 6.23021964751350246881882725271, 8.972504432680970898032209122777, 10.20458218628807267645251886245, 13.54710833338065349296574072216, 16.71303239842308479600166267384, 18.00274185120459235642841690396, 19.87494481669384388760742054607, 22.23571281374329264203523647512

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