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  − 5.05e13·2-s + 2.16e22·3-s − 7.34e27·4-s + 1.20e32·5-s − 1.09e36·6-s − 1.85e39·7-s + 8.72e41·8-s + 2.32e44·9-s − 6.07e45·10-s + 6.89e47·11-s − 1.58e50·12-s + 5.57e51·13-s + 9.36e52·14-s + 2.59e54·15-s + 2.86e55·16-s − 1.91e57·17-s − 1.17e58·18-s − 4.84e59·19-s − 8.82e59·20-s − 4.00e61·21-s − 3.48e61·22-s + 2.71e63·23-s + 1.88e64·24-s − 8.65e64·25-s − 2.82e65·26-s − 7.68e64·27-s + 1.36e67·28-s + ⋯
L(s)  = 1  − 0.508·2-s + 1.40·3-s − 0.741·4-s + 0.377·5-s − 0.716·6-s − 0.934·7-s + 0.885·8-s + 0.984·9-s − 0.192·10-s + 0.259·11-s − 1.04·12-s + 0.887·13-s + 0.474·14-s + 0.532·15-s + 0.291·16-s − 1.16·17-s − 0.500·18-s − 1.67·19-s − 0.280·20-s − 1.31·21-s − 0.131·22-s + 1.30·23-s + 1.24·24-s − 0.857·25-s − 0.450·26-s − 0.0212·27-s + 0.692·28-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 \( 1 + 5.05e13T + 9.90e27T^{2} \)
3 \( 1 - 2.16e22T + 2.35e44T^{2} \)
5 \( 1 - 1.20e32T + 1.00e65T^{2} \)
7 \( 1 + 1.85e39T + 3.92e78T^{2} \)
11 \( 1 - 6.89e47T + 7.07e96T^{2} \)
13 \( 1 - 5.57e51T + 3.95e103T^{2} \)
17 \( 1 + 1.91e57T + 2.70e114T^{2} \)
19 \( 1 + 4.84e59T + 8.39e118T^{2} \)
23 \( 1 - 2.71e63T + 4.37e126T^{2} \)
29 \( 1 - 1.52e68T + 1.00e136T^{2} \)
31 \( 1 - 1.67e69T + 4.97e138T^{2} \)
37 \( 1 + 8.48e72T + 6.96e145T^{2} \)
41 \( 1 + 1.65e75T + 9.74e149T^{2} \)
43 \( 1 + 1.12e75T + 8.17e151T^{2} \)
47 \( 1 - 5.85e76T + 3.19e155T^{2} \)
53 \( 1 + 1.38e79T + 2.27e160T^{2} \)
59 \( 1 + 4.69e81T + 4.88e164T^{2} \)
61 \( 1 + 1.85e83T + 1.08e166T^{2} \)
67 \( 1 + 7.55e84T + 6.68e169T^{2} \)
71 \( 1 - 1.37e86T + 1.46e172T^{2} \)
73 \( 1 + 4.41e86T + 1.94e173T^{2} \)
79 \( 1 + 3.34e88T + 3.01e176T^{2} \)
83 \( 1 + 4.06e88T + 2.98e178T^{2} \)
89 \( 1 - 2.67e89T + 1.96e181T^{2} \)
97 \( 1 + 5.40e91T + 5.88e184T^{2} \)
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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

−13.80633612930408281899628579818, −13.10038171611001927258730525854, −10.31064599630227469399190906316, −9.034514349718735408216006458533, −8.451777465815828646405332760904, −6.57606478474105690162043061189, −4.32304324376174698821086174483, −3.07628073342446657369249799081, −1.64982979947534669826343267758, 0, 1.64982979947534669826343267758, 3.07628073342446657369249799081, 4.32304324376174698821086174483, 6.57606478474105690162043061189, 8.451777465815828646405332760904, 9.034514349718735408216006458533, 10.31064599630227469399190906316, 13.10038171611001927258730525854, 13.80633612930408281899628579818

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