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 − 0.665·3-s + 4-s + 0.670·5-s + 0.665·6-s − 3.39·7-s − 8-s − 2.55·9-s − 0.670·10-s − 2.66·11-s − 0.665·12-s + 2.25·13-s + 3.39·14-s − 0.445·15-s + 16-s + 3.18·17-s + 2.55·18-s + 4.73·19-s + 0.670·20-s + 2.26·21-s + 2.66·22-s + 23-s + 0.665·24-s − 4.55·25-s − 2.25·26-s + 3.69·27-s − 3.39·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.384·3-s + 0.5·4-s + 0.299·5-s + 0.271·6-s − 1.28·7-s − 0.353·8-s − 0.852·9-s − 0.211·10-s − 0.803·11-s − 0.192·12-s + 0.624·13-s + 0.908·14-s − 0.115·15-s + 0.250·16-s + 0.772·17-s + 0.602·18-s + 1.08·19-s + 0.149·20-s + 0.493·21-s + 0.568·22-s + 0.208·23-s + 0.135·24-s − 0.910·25-s − 0.441·26-s + 0.711·27-s − 0.642·28-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 \( 1 + 0.665T + 3T^{2} \)
5 \( 1 - 0.670T + 5T^{2} \)
7 \( 1 + 3.39T + 7T^{2} \)
11 \( 1 + 2.66T + 11T^{2} \)
13 \( 1 - 2.25T + 13T^{2} \)
17 \( 1 - 3.18T + 17T^{2} \)
19 \( 1 - 4.73T + 19T^{2} \)
29 \( 1 + 9.07T + 29T^{2} \)
31 \( 1 - 2.40T + 31T^{2} \)
37 \( 1 - 10.0T + 37T^{2} \)
41 \( 1 - 10.3T + 41T^{2} \)
43 \( 1 + 10.8T + 43T^{2} \)
47 \( 1 - 2.12T + 47T^{2} \)
53 \( 1 - 2.74T + 53T^{2} \)
59 \( 1 - 2.19T + 59T^{2} \)
61 \( 1 - 0.869T + 61T^{2} \)
67 \( 1 + 5.55T + 67T^{2} \)
71 \( 1 - 13.2T + 71T^{2} \)
73 \( 1 - 0.381T + 73T^{2} \)
79 \( 1 + 6.31T + 79T^{2} \)
83 \( 1 + 8.49T + 83T^{2} \)
89 \( 1 - 13.3T + 89T^{2} \)
97 \( 1 - 3.23T + 97T^{2} \)
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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

−7.76541334177531282021311305449, −7.10852275073400326911689895400, −6.09028163465065179085006678613, −5.88963934044936981753082582354, −5.16495485050462166271230042744, −3.76919340758210988829741513474, −3.09336593746800491337126625862, −2.37318711458875102805469288779, −1.01967887789016260037552235534, 0, 1.01967887789016260037552235534, 2.37318711458875102805469288779, 3.09336593746800491337126625862, 3.76919340758210988829741513474, 5.16495485050462166271230042744, 5.88963934044936981753082582354, 6.09028163465065179085006678613, 7.10852275073400326911689895400, 7.76541334177531282021311305449

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