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 − 2.41·3-s + 4-s + 0.460·5-s + 2.41·6-s − 4.49·7-s − 8-s + 2.81·9-s − 0.460·10-s − 2.62·11-s − 2.41·12-s + 0.877·13-s + 4.49·14-s − 1.11·15-s + 16-s + 0.794·17-s − 2.81·18-s + 0.648·19-s + 0.460·20-s + 10.8·21-s + 2.62·22-s − 23-s + 2.41·24-s − 4.78·25-s − 0.877·26-s + 0.436·27-s − 4.49·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.39·3-s + 0.5·4-s + 0.205·5-s + 0.984·6-s − 1.69·7-s − 0.353·8-s + 0.939·9-s − 0.145·10-s − 0.791·11-s − 0.696·12-s + 0.243·13-s + 1.20·14-s − 0.286·15-s + 0.250·16-s + 0.192·17-s − 0.664·18-s + 0.148·19-s + 0.102·20-s + 2.36·21-s + 0.559·22-s − 0.208·23-s + 0.492·24-s − 0.957·25-s − 0.172·26-s + 0.0840·27-s − 0.849·28-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 \( 1 + 2.41T + 3T^{2} \)
5 \( 1 - 0.460T + 5T^{2} \)
7 \( 1 + 4.49T + 7T^{2} \)
11 \( 1 + 2.62T + 11T^{2} \)
13 \( 1 - 0.877T + 13T^{2} \)
17 \( 1 - 0.794T + 17T^{2} \)
19 \( 1 - 0.648T + 19T^{2} \)
29 \( 1 - 2.58T + 29T^{2} \)
31 \( 1 + 6.97T + 31T^{2} \)
37 \( 1 - 7.43T + 37T^{2} \)
41 \( 1 - 1.59T + 41T^{2} \)
43 \( 1 + 1.00T + 43T^{2} \)
47 \( 1 + 8.71T + 47T^{2} \)
53 \( 1 - 8.84T + 53T^{2} \)
59 \( 1 - 3.77T + 59T^{2} \)
61 \( 1 - 3.78T + 61T^{2} \)
67 \( 1 - 14.9T + 67T^{2} \)
71 \( 1 + 6.81T + 71T^{2} \)
73 \( 1 - 8.09T + 73T^{2} \)
79 \( 1 + 11.8T + 79T^{2} \)
83 \( 1 - 14.8T + 83T^{2} \)
89 \( 1 + 9.69T + 89T^{2} \)
97 \( 1 + 3.98T + 97T^{2} \)
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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

−7.58051015874970773829677862955, −6.87430528840611900928917447800, −6.29258853321209574642620713486, −5.77700848041647279210334846982, −5.20337501325891763019288981117, −4.00350930912498325921334072281, −3.14493725566329576952007462657, −2.20392439175238487671007047517, −0.810484223941480564438502695978, 0, 0.810484223941480564438502695978, 2.20392439175238487671007047517, 3.14493725566329576952007462657, 4.00350930912498325921334072281, 5.20337501325891763019288981117, 5.77700848041647279210334846982, 6.29258853321209574642620713486, 6.87430528840611900928917447800, 7.58051015874970773829677862955

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