Properties

Degree $2$
Conductor $1$
Sign $-1$
Motivic weight $33$
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4.96e4·2-s − 1.55e7·3-s − 6.12e9·4-s + 2.04e10·5-s − 7.71e11·6-s − 1.22e14·7-s − 7.30e14·8-s − 5.31e15·9-s + 1.01e15·10-s + 2.15e17·11-s + 9.50e16·12-s − 1.07e18·13-s − 6.06e18·14-s − 3.17e17·15-s + 1.62e19·16-s + 2.54e20·17-s − 2.64e20·18-s − 1.22e21·19-s − 1.25e20·20-s + 1.89e21·21-s + 1.07e22·22-s − 5.11e21·23-s + 1.13e22·24-s − 1.15e23·25-s − 5.35e22·26-s + 1.68e23·27-s + 7.47e23·28-s + ⋯
L(s)  = 1  + 0.536·2-s − 0.208·3-s − 0.712·4-s + 0.0598·5-s − 0.111·6-s − 1.38·7-s − 0.918·8-s − 0.956·9-s + 0.0320·10-s + 1.41·11-s + 0.148·12-s − 0.449·13-s − 0.744·14-s − 0.0124·15-s + 0.220·16-s + 1.26·17-s − 0.512·18-s − 0.970·19-s − 0.0426·20-s + 0.289·21-s + 0.758·22-s − 0.173·23-s + 0.191·24-s − 0.996·25-s − 0.240·26-s + 0.407·27-s + 0.990·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & -\,\Lambda(34-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+33/2) \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1\)
Sign: $-1$
Motivic weight: \(33\)
Character: $\chi_{1} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1,\ (\ :33/2),\ -1)\)

Particular Values

\(L(17)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{35}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
good2 \( 1 - 4.96e4T + 8.58e9T^{2} \)
3 \( 1 + 1.55e7T + 5.55e15T^{2} \)
5 \( 1 - 2.04e10T + 1.16e23T^{2} \)
7 \( 1 + 1.22e14T + 7.73e27T^{2} \)
11 \( 1 - 2.15e17T + 2.32e34T^{2} \)
13 \( 1 + 1.07e18T + 5.75e36T^{2} \)
17 \( 1 - 2.54e20T + 4.02e40T^{2} \)
19 \( 1 + 1.22e21T + 1.58e42T^{2} \)
23 \( 1 + 5.11e21T + 8.65e44T^{2} \)
29 \( 1 + 1.64e23T + 1.81e48T^{2} \)
31 \( 1 + 6.75e24T + 1.64e49T^{2} \)
37 \( 1 + 7.46e25T + 5.63e51T^{2} \)
41 \( 1 - 4.96e26T + 1.66e53T^{2} \)
43 \( 1 + 1.99e26T + 8.02e53T^{2} \)
47 \( 1 - 2.16e27T + 1.51e55T^{2} \)
53 \( 1 + 3.60e28T + 7.96e56T^{2} \)
59 \( 1 + 1.87e29T + 2.74e58T^{2} \)
61 \( 1 - 4.18e27T + 8.23e58T^{2} \)
67 \( 1 - 4.85e29T + 1.82e60T^{2} \)
71 \( 1 + 3.42e30T + 1.23e61T^{2} \)
73 \( 1 - 7.01e30T + 3.08e61T^{2} \)
79 \( 1 + 2.95e30T + 4.18e62T^{2} \)
83 \( 1 + 1.23e31T + 2.13e63T^{2} \)
89 \( 1 - 7.05e31T + 2.13e64T^{2} \)
97 \( 1 + 7.71e32T + 3.65e65T^{2} \)
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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

−23.17164710223873043569343353110, −22.08212852190947044930996024239, −19.37737814171362848300641710869, −16.99729322981038600458338655937, −14.33154534246216629846064850558, −12.38824383391611598453844590283, −9.337420969687700231085444375144, −5.97786009657894493138722886448, −3.56400685368044213449625106689, 0, 3.56400685368044213449625106689, 5.97786009657894493138722886448, 9.337420969687700231085444375144, 12.38824383391611598453844590283, 14.33154534246216629846064850558, 16.99729322981038600458338655937, 19.37737814171362848300641710869, 22.08212852190947044930996024239, 23.17164710223873043569343353110

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