Properties

Degree 2
Conductor $ 2 \cdot 23 \cdot 131 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 0.785·3-s + 4-s − 1.42·5-s + 0.785·6-s + 3.87·7-s − 8-s − 2.38·9-s + 1.42·10-s + 4.24·11-s − 0.785·12-s + 2.46·13-s − 3.87·14-s + 1.11·15-s + 16-s + 1.55·17-s + 2.38·18-s − 2.53·19-s − 1.42·20-s − 3.04·21-s − 4.24·22-s − 23-s + 0.785·24-s − 2.97·25-s − 2.46·26-s + 4.22·27-s + 3.87·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.453·3-s + 0.5·4-s − 0.636·5-s + 0.320·6-s + 1.46·7-s − 0.353·8-s − 0.794·9-s + 0.449·10-s + 1.28·11-s − 0.226·12-s + 0.684·13-s − 1.03·14-s + 0.288·15-s + 0.250·16-s + 0.376·17-s + 0.561·18-s − 0.581·19-s − 0.318·20-s − 0.664·21-s − 0.905·22-s − 0.208·23-s + 0.160·24-s − 0.595·25-s − 0.483·26-s + 0.813·27-s + 0.732·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6026\)    =    \(2 \cdot 23 \cdot 131\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{6026} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 6026,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;23,\;131\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;23,\;131\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 + T \)
23 \( 1 + T \)
131 \( 1 + T \)
good3 \( 1 + 0.785T + 3T^{2} \)
5 \( 1 + 1.42T + 5T^{2} \)
7 \( 1 - 3.87T + 7T^{2} \)
11 \( 1 - 4.24T + 11T^{2} \)
13 \( 1 - 2.46T + 13T^{2} \)
17 \( 1 - 1.55T + 17T^{2} \)
19 \( 1 + 2.53T + 19T^{2} \)
29 \( 1 - 4.09T + 29T^{2} \)
31 \( 1 + 7.27T + 31T^{2} \)
37 \( 1 + 5.04T + 37T^{2} \)
41 \( 1 + 6.34T + 41T^{2} \)
43 \( 1 + 9.30T + 43T^{2} \)
47 \( 1 + 5.01T + 47T^{2} \)
53 \( 1 + 6.61T + 53T^{2} \)
59 \( 1 + 1.17T + 59T^{2} \)
61 \( 1 - 6.46T + 61T^{2} \)
67 \( 1 - 9.88T + 67T^{2} \)
71 \( 1 - 6.27T + 71T^{2} \)
73 \( 1 + 13.0T + 73T^{2} \)
79 \( 1 + 8.91T + 79T^{2} \)
83 \( 1 - 14.1T + 83T^{2} \)
89 \( 1 + 11.5T + 89T^{2} \)
97 \( 1 + 14.5T + 97T^{2} \)
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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

−8.025789971808667408500548922619, −7.02514469904903957391448443901, −6.45755012974403635298279615357, −5.61657805371974909869042490042, −4.92069859225941676131467131883, −3.99643880645125027113445767435, −3.30376462166829058128289419170, −1.92061952969320063084201513158, −1.28773358632734674168327701468, 0, 1.28773358632734674168327701468, 1.92061952969320063084201513158, 3.30376462166829058128289419170, 3.99643880645125027113445767435, 4.92069859225941676131467131883, 5.61657805371974909869042490042, 6.45755012974403635298279615357, 7.02514469904903957391448443901, 8.025789971808667408500548922619

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