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  − 1.67e14·2-s + 1.22e22·3-s + 1.80e28·4-s − 5.69e32·5-s − 2.05e36·6-s + 3.48e39·7-s − 1.36e42·8-s − 8.52e43·9-s + 9.52e46·10-s − 4.56e47·11-s + 2.21e50·12-s − 4.65e49·13-s − 5.82e53·14-s − 6.98e54·15-s + 4.92e55·16-s − 7.16e56·17-s + 1.42e58·18-s − 4.92e58·19-s − 1.02e61·20-s + 4.27e61·21-s + 7.64e61·22-s + 1.15e63·23-s − 1.67e64·24-s + 2.23e65·25-s + 7.78e63·26-s − 3.93e66·27-s + 6.29e67·28-s + ⋯
L(s)  = 1  − 1.68·2-s + 0.798·3-s + 1.82·4-s − 1.79·5-s − 1.34·6-s + 1.75·7-s − 1.38·8-s − 0.361·9-s + 3.01·10-s − 0.171·11-s + 1.45·12-s − 0.00740·13-s − 2.95·14-s − 1.43·15-s + 0.502·16-s − 0.435·17-s + 0.607·18-s − 0.170·19-s − 3.26·20-s + 1.40·21-s + 0.288·22-s + 0.553·23-s − 1.10·24-s + 2.21·25-s + 0.0124·26-s − 1.08·27-s + 3.20·28-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

\( d \)  =  \(2\)
\( N \)  =  \(1\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(93\)
character  :  $\chi_{1} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(1\)
Selberg data  =  \((2,\ 1,\ (\ :93/2),\ -1)\)
\(L(47)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{95}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where,\(F_p(T)\) is a polynomial of degree 2.
$p$$F_p(T)$
good2 \( 1 + 1.67e14T + 9.90e27T^{2} \)
3 \( 1 - 1.22e22T + 2.35e44T^{2} \)
5 \( 1 + 5.69e32T + 1.00e65T^{2} \)
7 \( 1 - 3.48e39T + 3.92e78T^{2} \)
11 \( 1 + 4.56e47T + 7.07e96T^{2} \)
13 \( 1 + 4.65e49T + 3.95e103T^{2} \)
17 \( 1 + 7.16e56T + 2.70e114T^{2} \)
19 \( 1 + 4.92e58T + 8.39e118T^{2} \)
23 \( 1 - 1.15e63T + 4.37e126T^{2} \)
29 \( 1 - 1.03e68T + 1.00e136T^{2} \)
31 \( 1 + 8.93e67T + 4.97e138T^{2} \)
37 \( 1 - 5.91e72T + 6.96e145T^{2} \)
41 \( 1 - 2.38e74T + 9.74e149T^{2} \)
43 \( 1 + 1.84e75T + 8.17e151T^{2} \)
47 \( 1 - 5.58e77T + 3.19e155T^{2} \)
53 \( 1 + 2.41e80T + 2.27e160T^{2} \)
59 \( 1 + 2.30e82T + 4.88e164T^{2} \)
61 \( 1 - 1.46e83T + 1.08e166T^{2} \)
67 \( 1 - 1.40e84T + 6.68e169T^{2} \)
71 \( 1 + 1.74e86T + 1.46e172T^{2} \)
73 \( 1 + 4.65e86T + 1.94e173T^{2} \)
79 \( 1 + 1.89e88T + 3.01e176T^{2} \)
83 \( 1 + 2.23e89T + 2.98e178T^{2} \)
89 \( 1 + 1.44e90T + 1.96e181T^{2} \)
97 \( 1 + 8.44e91T + 5.88e184T^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−14.62270726657566862494143753183, −11.66063967226511547486506927556, −10.92027683044117204945125492273, −8.745577242447109061775594974382, −8.151138666962428082219685898071, −7.39077128598606968431656629666, −4.44286336617295541313490891036, −2.69238081254357342412253396197, −1.24799338765042645218901883671, 0, 1.24799338765042645218901883671, 2.69238081254357342412253396197, 4.44286336617295541313490891036, 7.39077128598606968431656629666, 8.151138666962428082219685898071, 8.745577242447109061775594974382, 10.92027683044117204945125492273, 11.66063967226511547486506927556, 14.62270726657566862494143753183

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