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 + 2.63·3-s + 4-s − 1.57·5-s − 2.63·6-s − 2.35·7-s − 8-s + 3.93·9-s + 1.57·10-s + 2.96·11-s + 2.63·12-s + 2.20·13-s + 2.35·14-s − 4.14·15-s + 16-s − 0.149·17-s − 3.93·18-s − 4.54·19-s − 1.57·20-s − 6.19·21-s − 2.96·22-s + 23-s − 2.63·24-s − 2.52·25-s − 2.20·26-s + 2.47·27-s − 2.35·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.52·3-s + 0.5·4-s − 0.703·5-s − 1.07·6-s − 0.888·7-s − 0.353·8-s + 1.31·9-s + 0.497·10-s + 0.894·11-s + 0.760·12-s + 0.611·13-s + 0.628·14-s − 1.06·15-s + 0.250·16-s − 0.0362·17-s − 0.928·18-s − 1.04·19-s − 0.351·20-s − 1.35·21-s − 0.632·22-s + 0.208·23-s − 0.537·24-s − 0.505·25-s − 0.432·26-s + 0.475·27-s − 0.444·28-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 \( 1 - 2.63T + 3T^{2} \)
5 \( 1 + 1.57T + 5T^{2} \)
7 \( 1 + 2.35T + 7T^{2} \)
11 \( 1 - 2.96T + 11T^{2} \)
13 \( 1 - 2.20T + 13T^{2} \)
17 \( 1 + 0.149T + 17T^{2} \)
19 \( 1 + 4.54T + 19T^{2} \)
29 \( 1 + 2.19T + 29T^{2} \)
31 \( 1 - 4.09T + 31T^{2} \)
37 \( 1 + 10.6T + 37T^{2} \)
41 \( 1 + 9.66T + 41T^{2} \)
43 \( 1 - 7.19T + 43T^{2} \)
47 \( 1 + 10.4T + 47T^{2} \)
53 \( 1 - 4.11T + 53T^{2} \)
59 \( 1 + 1.51T + 59T^{2} \)
61 \( 1 - 8.49T + 61T^{2} \)
67 \( 1 + 3.65T + 67T^{2} \)
71 \( 1 - 10.0T + 71T^{2} \)
73 \( 1 + 9.06T + 73T^{2} \)
79 \( 1 - 3.04T + 79T^{2} \)
83 \( 1 + 4.50T + 83T^{2} \)
89 \( 1 + 7.58T + 89T^{2} \)
97 \( 1 + 6.63T + 97T^{2} \)
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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

−8.067153392157615364658398565741, −7.04206066536716140039549484345, −6.75148455594140969576808769217, −5.83778893391511023968610941470, −4.45888202643597090457311292478, −3.59760632577411631138183631181, −3.37428724303693409102492053975, −2.30645477896050177829763866439, −1.46274412810410308467676941216, 0, 1.46274412810410308467676941216, 2.30645477896050177829763866439, 3.37428724303693409102492053975, 3.59760632577411631138183631181, 4.45888202643597090457311292478, 5.83778893391511023968610941470, 6.75148455594140969576808769217, 7.04206066536716140039549484345, 8.067153392157615364658398565741

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