Properties

Degree $2$
Conductor $1$
Sign $-1$
Motivic weight $101$
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5.93e14·2-s + 2.15e24·3-s − 2.18e30·4-s + 2.07e34·5-s + 1.27e39·6-s − 6.15e42·7-s − 2.80e45·8-s + 3.07e48·9-s + 1.23e49·10-s − 1.93e51·11-s − 4.69e54·12-s + 2.07e56·13-s − 3.65e57·14-s + 4.46e58·15-s + 3.87e60·16-s + 9.51e61·17-s + 1.82e63·18-s − 2.07e64·19-s − 4.53e64·20-s − 1.32e67·21-s − 1.14e66·22-s − 1.01e69·23-s − 6.02e69·24-s − 3.90e70·25-s + 1.23e71·26-s + 3.29e72·27-s + 1.34e73·28-s + ⋯
L(s)  = 1  + 0.372·2-s + 1.72·3-s − 0.861·4-s + 0.104·5-s + 0.644·6-s − 1.29·7-s − 0.693·8-s + 1.99·9-s + 0.0389·10-s − 0.0497·11-s − 1.48·12-s + 1.15·13-s − 0.482·14-s + 0.180·15-s + 0.602·16-s + 0.692·17-s + 0.742·18-s − 0.549·19-s − 0.0900·20-s − 2.23·21-s − 0.0185·22-s − 1.72·23-s − 1.19·24-s − 0.989·25-s + 0.431·26-s + 1.71·27-s + 1.11·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & -\,\Lambda(102-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+50.5) \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1\)
Sign: $-1$
Motivic weight: \(101\)
Character: $\chi_{1} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1,\ (\ :101/2),\ -1)\)

Particular Values

\(L(51)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{103}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
good2 \( 1 - 5.93e14T + 2.53e30T^{2} \)
3 \( 1 - 2.15e24T + 1.54e48T^{2} \)
5 \( 1 - 2.07e34T + 3.94e70T^{2} \)
7 \( 1 + 6.15e42T + 2.26e85T^{2} \)
11 \( 1 + 1.93e51T + 1.51e105T^{2} \)
13 \( 1 - 2.07e56T + 3.22e112T^{2} \)
17 \( 1 - 9.51e61T + 1.88e124T^{2} \)
19 \( 1 + 2.07e64T + 1.42e129T^{2} \)
23 \( 1 + 1.01e69T + 3.42e137T^{2} \)
29 \( 1 + 1.12e74T + 5.03e147T^{2} \)
31 \( 1 + 3.39e75T + 4.24e150T^{2} \)
37 \( 1 + 2.80e78T + 2.44e158T^{2} \)
41 \( 1 + 2.18e81T + 7.78e162T^{2} \)
43 \( 1 - 4.27e82T + 9.55e164T^{2} \)
47 \( 1 + 4.06e84T + 7.61e168T^{2} \)
53 \( 1 + 1.28e87T + 1.41e174T^{2} \)
59 \( 1 - 2.29e88T + 7.17e178T^{2} \)
61 \( 1 - 1.48e90T + 2.08e180T^{2} \)
67 \( 1 + 1.12e92T + 2.71e184T^{2} \)
71 \( 1 + 1.07e93T + 9.48e186T^{2} \)
73 \( 1 - 8.88e92T + 1.56e188T^{2} \)
79 \( 1 - 3.94e95T + 4.57e191T^{2} \)
83 \( 1 - 7.91e96T + 6.71e193T^{2} \)
89 \( 1 - 1.56e98T + 7.73e196T^{2} \)
97 \( 1 - 6.75e99T + 4.61e200T^{2} \)
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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

−13.53777016020955080219130243978, −12.79034750111152199349906633243, −9.876976691829167871298209006187, −9.064160490326393004639494130060, −7.87224993132106916662924101139, −5.98121580683759678990142391321, −3.80628493334499079304770050752, −3.46578536123029202680226215033, −1.88185247019151326302359454216, 0, 1.88185247019151326302359454216, 3.46578536123029202680226215033, 3.80628493334499079304770050752, 5.98121580683759678990142391321, 7.87224993132106916662924101139, 9.064160490326393004639494130060, 9.876976691829167871298209006187, 12.79034750111152199349906633243, 13.53777016020955080219130243978

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