Properties

Degree $2$
Conductor $1$
Sign $-1$
Motivic weight $93$
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 8.17e13·2-s + 3.94e21·3-s − 3.22e27·4-s + 2.56e32·5-s + 3.22e35·6-s + 2.42e39·7-s − 1.07e42·8-s − 2.20e44·9-s + 2.09e46·10-s − 3.34e48·11-s − 1.27e49·12-s − 2.46e51·13-s + 1.98e53·14-s + 1.01e54·15-s − 5.56e55·16-s + 8.99e56·17-s − 1.79e58·18-s + 1.17e58·19-s − 8.27e59·20-s + 9.59e60·21-s − 2.73e62·22-s + 2.35e63·23-s − 4.23e63·24-s − 3.52e64·25-s − 2.01e65·26-s − 1.79e66·27-s − 7.84e66·28-s + ⋯
L(s)  = 1  + 0.820·2-s + 0.257·3-s − 0.325·4-s + 0.806·5-s + 0.211·6-s + 1.22·7-s − 1.08·8-s − 0.933·9-s + 0.662·10-s − 1.25·11-s − 0.0838·12-s − 0.392·13-s + 1.00·14-s + 0.207·15-s − 0.567·16-s + 0.546·17-s − 0.766·18-s + 0.0406·19-s − 0.262·20-s + 0.315·21-s − 1.03·22-s + 1.12·23-s − 0.280·24-s − 0.349·25-s − 0.322·26-s − 0.497·27-s − 0.399·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & -\,\Lambda(94-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+93/2) \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1\)
Sign: $-1$
Motivic weight: \(93\)
Character: $\chi_{1} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1,\ (\ :93/2),\ -1)\)

Particular Values

\(L(47)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{95}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
good2 \( 1 - 8.17e13T + 9.90e27T^{2} \)
3 \( 1 - 3.94e21T + 2.35e44T^{2} \)
5 \( 1 - 2.56e32T + 1.00e65T^{2} \)
7 \( 1 - 2.42e39T + 3.92e78T^{2} \)
11 \( 1 + 3.34e48T + 7.07e96T^{2} \)
13 \( 1 + 2.46e51T + 3.95e103T^{2} \)
17 \( 1 - 8.99e56T + 2.70e114T^{2} \)
19 \( 1 - 1.17e58T + 8.39e118T^{2} \)
23 \( 1 - 2.35e63T + 4.37e126T^{2} \)
29 \( 1 + 1.13e68T + 1.00e136T^{2} \)
31 \( 1 + 4.06e69T + 4.97e138T^{2} \)
37 \( 1 + 3.61e72T + 6.96e145T^{2} \)
41 \( 1 + 1.25e74T + 9.74e149T^{2} \)
43 \( 1 - 6.73e75T + 8.17e151T^{2} \)
47 \( 1 + 9.64e77T + 3.19e155T^{2} \)
53 \( 1 + 1.68e80T + 2.27e160T^{2} \)
59 \( 1 + 1.86e82T + 4.88e164T^{2} \)
61 \( 1 + 1.57e83T + 1.08e166T^{2} \)
67 \( 1 - 1.27e85T + 6.68e169T^{2} \)
71 \( 1 - 6.05e84T + 1.46e172T^{2} \)
73 \( 1 - 5.11e86T + 1.94e173T^{2} \)
79 \( 1 + 2.35e88T + 3.01e176T^{2} \)
83 \( 1 + 1.36e88T + 2.98e178T^{2} \)
89 \( 1 - 5.78e90T + 1.96e181T^{2} \)
97 \( 1 + 1.91e92T + 5.88e184T^{2} \)
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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

−14.06359508468588970785217311747, −12.86313068279808802504668495513, −11.12390484434279658313745324356, −9.292268900487729938822699030291, −7.900348453669298087081846545855, −5.61204218607483688429891194675, −5.00516304517308193266802192254, −3.19749320409605204402321686184, −1.93543058059972283252551390914, 0, 1.93543058059972283252551390914, 3.19749320409605204402321686184, 5.00516304517308193266802192254, 5.61204218607483688429891194675, 7.900348453669298087081846545855, 9.292268900487729938822699030291, 11.12390484434279658313745324356, 12.86313068279808802504668495513, 14.06359508468588970785217311747

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