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 + 2.57·3-s + 4-s + 3.18·5-s − 2.57·6-s − 4.63·7-s − 8-s + 3.65·9-s − 3.18·10-s − 1.26·11-s + 2.57·12-s − 4.01·13-s + 4.63·14-s + 8.20·15-s + 16-s + 1.91·17-s − 3.65·18-s − 4.81·19-s + 3.18·20-s − 11.9·21-s + 1.26·22-s + 23-s − 2.57·24-s + 5.12·25-s + 4.01·26-s + 1.68·27-s − 4.63·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.48·3-s + 0.5·4-s + 1.42·5-s − 1.05·6-s − 1.75·7-s − 0.353·8-s + 1.21·9-s − 1.00·10-s − 0.381·11-s + 0.744·12-s − 1.11·13-s + 1.23·14-s + 2.11·15-s + 0.250·16-s + 0.465·17-s − 0.860·18-s − 1.10·19-s + 0.711·20-s − 2.60·21-s + 0.269·22-s + 0.208·23-s − 0.526·24-s + 1.02·25-s + 0.787·26-s + 0.323·27-s − 0.876·28-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 \( 1 - 2.57T + 3T^{2} \)
5 \( 1 - 3.18T + 5T^{2} \)
7 \( 1 + 4.63T + 7T^{2} \)
11 \( 1 + 1.26T + 11T^{2} \)
13 \( 1 + 4.01T + 13T^{2} \)
17 \( 1 - 1.91T + 17T^{2} \)
19 \( 1 + 4.81T + 19T^{2} \)
29 \( 1 - 2.52T + 29T^{2} \)
31 \( 1 - 0.522T + 31T^{2} \)
37 \( 1 - 0.765T + 37T^{2} \)
41 \( 1 + 3.04T + 41T^{2} \)
43 \( 1 + 12.2T + 43T^{2} \)
47 \( 1 + 4.23T + 47T^{2} \)
53 \( 1 - 2.15T + 53T^{2} \)
59 \( 1 + 12.2T + 59T^{2} \)
61 \( 1 - 7.28T + 61T^{2} \)
67 \( 1 - 0.101T + 67T^{2} \)
71 \( 1 + 12.4T + 71T^{2} \)
73 \( 1 - 7.43T + 73T^{2} \)
79 \( 1 + 8.28T + 79T^{2} \)
83 \( 1 + 12.4T + 83T^{2} \)
89 \( 1 - 12.0T + 89T^{2} \)
97 \( 1 - 4.39T + 97T^{2} \)
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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

−7.85535660873221889518468194719, −7.05539417803620660269034714086, −6.50614240632999307048938116909, −5.85890388827336730180332540744, −4.84959307689045157222316510157, −3.59133165268862119375145523672, −2.87398660836735175519754740443, −2.45207833151307921265963047858, −1.63591772496228960282424887750, 0, 1.63591772496228960282424887750, 2.45207833151307921265963047858, 2.87398660836735175519754740443, 3.59133165268862119375145523672, 4.84959307689045157222316510157, 5.85890388827336730180332540744, 6.50614240632999307048938116909, 7.05539417803620660269034714086, 7.85535660873221889518468194719

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