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 + 0.694·3-s + 4-s + 0.658·5-s − 0.694·6-s − 2.75·7-s − 8-s − 2.51·9-s − 0.658·10-s − 3.14·11-s + 0.694·12-s − 1.49·13-s + 2.75·14-s + 0.457·15-s + 16-s + 7.27·17-s + 2.51·18-s + 5.41·19-s + 0.658·20-s − 1.91·21-s + 3.14·22-s + 23-s − 0.694·24-s − 4.56·25-s + 1.49·26-s − 3.83·27-s − 2.75·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.401·3-s + 0.5·4-s + 0.294·5-s − 0.283·6-s − 1.03·7-s − 0.353·8-s − 0.839·9-s − 0.208·10-s − 0.949·11-s + 0.200·12-s − 0.414·13-s + 0.735·14-s + 0.118·15-s + 0.250·16-s + 1.76·17-s + 0.593·18-s + 1.24·19-s + 0.147·20-s − 0.416·21-s + 0.671·22-s + 0.208·23-s − 0.141·24-s − 0.913·25-s + 0.293·26-s − 0.737·27-s − 0.519·28-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 \( 1 - 0.694T + 3T^{2} \)
5 \( 1 - 0.658T + 5T^{2} \)
7 \( 1 + 2.75T + 7T^{2} \)
11 \( 1 + 3.14T + 11T^{2} \)
13 \( 1 + 1.49T + 13T^{2} \)
17 \( 1 - 7.27T + 17T^{2} \)
19 \( 1 - 5.41T + 19T^{2} \)
29 \( 1 - 7.88T + 29T^{2} \)
31 \( 1 + 1.13T + 31T^{2} \)
37 \( 1 + 0.745T + 37T^{2} \)
41 \( 1 - 0.571T + 41T^{2} \)
43 \( 1 - 5.56T + 43T^{2} \)
47 \( 1 - 0.938T + 47T^{2} \)
53 \( 1 + 10.5T + 53T^{2} \)
59 \( 1 + 6.71T + 59T^{2} \)
61 \( 1 - 12.7T + 61T^{2} \)
67 \( 1 - 6.08T + 67T^{2} \)
71 \( 1 + 8.32T + 71T^{2} \)
73 \( 1 - 8.52T + 73T^{2} \)
79 \( 1 - 0.159T + 79T^{2} \)
83 \( 1 + 1.60T + 83T^{2} \)
89 \( 1 + 15.4T + 89T^{2} \)
97 \( 1 + 9.93T + 97T^{2} \)
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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

−7.86050845200888932435798474056, −7.24504638429394658635724079208, −6.32870702315100120524341281105, −5.64243009988081951865815607734, −5.13588164922777495247338937524, −3.66032949780723688649622745322, −2.96829532819926669872050973647, −2.54462854843300157548266948676, −1.17941779297711817050049152493, 0, 1.17941779297711817050049152493, 2.54462854843300157548266948676, 2.96829532819926669872050973647, 3.66032949780723688649622745322, 5.13588164922777495247338937524, 5.64243009988081951865815607734, 6.32870702315100120524341281105, 7.24504638429394658635724079208, 7.86050845200888932435798474056

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