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  + 1.64e14·2-s − 2.77e22·3-s + 1.71e28·4-s + 3.18e32·5-s − 4.56e36·6-s − 2.05e39·7-s + 1.19e42·8-s + 5.33e44·9-s + 5.24e46·10-s − 1.16e48·11-s − 4.75e50·12-s + 9.03e51·13-s − 3.37e53·14-s − 8.84e54·15-s + 2.60e55·16-s + 2.65e56·17-s + 8.78e58·18-s − 2.80e59·19-s + 5.46e60·20-s + 5.69e61·21-s − 1.92e62·22-s − 1.45e63·23-s − 3.30e64·24-s + 7.21e62·25-s + 1.48e66·26-s − 8.27e66·27-s − 3.52e67·28-s + ⋯
L(s)  = 1  + 1.65·2-s − 1.80·3-s + 1.73·4-s + 1.00·5-s − 2.98·6-s − 1.03·7-s + 1.20·8-s + 2.26·9-s + 1.65·10-s − 0.439·11-s − 3.12·12-s + 1.43·13-s − 1.71·14-s − 1.81·15-s + 0.265·16-s + 0.161·17-s + 3.74·18-s − 0.967·19-s + 1.73·20-s + 1.87·21-s − 0.725·22-s − 0.696·23-s − 2.18·24-s + 0.00714·25-s + 2.37·26-s − 2.28·27-s − 1.79·28-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 \( 1 - 1.64e14T + 9.90e27T^{2} \)
3 \( 1 + 2.77e22T + 2.35e44T^{2} \)
5 \( 1 - 3.18e32T + 1.00e65T^{2} \)
7 \( 1 + 2.05e39T + 3.92e78T^{2} \)
11 \( 1 + 1.16e48T + 7.07e96T^{2} \)
13 \( 1 - 9.03e51T + 3.95e103T^{2} \)
17 \( 1 - 2.65e56T + 2.70e114T^{2} \)
19 \( 1 + 2.80e59T + 8.39e118T^{2} \)
23 \( 1 + 1.45e63T + 4.37e126T^{2} \)
29 \( 1 + 8.91e67T + 1.00e136T^{2} \)
31 \( 1 - 8.50e68T + 4.97e138T^{2} \)
37 \( 1 - 2.81e72T + 6.96e145T^{2} \)
41 \( 1 + 1.66e75T + 9.74e149T^{2} \)
43 \( 1 + 6.95e75T + 8.17e151T^{2} \)
47 \( 1 + 7.32e77T + 3.19e155T^{2} \)
53 \( 1 + 1.26e80T + 2.27e160T^{2} \)
59 \( 1 - 2.67e81T + 4.88e164T^{2} \)
61 \( 1 + 1.91e82T + 1.08e166T^{2} \)
67 \( 1 - 1.94e84T + 6.68e169T^{2} \)
71 \( 1 + 1.09e85T + 1.46e172T^{2} \)
73 \( 1 - 3.20e86T + 1.94e173T^{2} \)
79 \( 1 - 3.00e88T + 3.01e176T^{2} \)
83 \( 1 + 2.46e89T + 2.98e178T^{2} \)
89 \( 1 + 2.79e89T + 1.96e181T^{2} \)
97 \( 1 - 3.53e92T + 5.88e184T^{2} \)
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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

−13.43510357786557044316155537896, −12.74982508298504769612649045502, −11.36527660445730902433068074702, −10.12619205911426526469341229663, −6.42647963894505932033272995503, −6.12719171466360395466622317663, −5.05429458047663572858791949171, −3.63339109649557768810454088784, −1.76598384822919589294929158602, 0, 1.76598384822919589294929158602, 3.63339109649557768810454088784, 5.05429458047663572858791949171, 6.12719171466360395466622317663, 6.42647963894505932033272995503, 10.12619205911426526469341229663, 11.36527660445730902433068074702, 12.74982508298504769612649045502, 13.43510357786557044316155537896

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