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 − 0.510·3-s + 4-s + 2.36·5-s + 0.510·6-s + 2.78·7-s − 8-s − 2.73·9-s − 2.36·10-s + 0.898·11-s − 0.510·12-s + 1.36·13-s − 2.78·14-s − 1.21·15-s + 16-s − 4.57·17-s + 2.73·18-s + 3.53·19-s + 2.36·20-s − 1.42·21-s − 0.898·22-s − 23-s + 0.510·24-s + 0.613·25-s − 1.36·26-s + 2.93·27-s + 2.78·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.295·3-s + 0.5·4-s + 1.05·5-s + 0.208·6-s + 1.05·7-s − 0.353·8-s − 0.912·9-s − 0.749·10-s + 0.270·11-s − 0.147·12-s + 0.377·13-s − 0.745·14-s − 0.312·15-s + 0.250·16-s − 1.11·17-s + 0.645·18-s + 0.811·19-s + 0.529·20-s − 0.310·21-s − 0.191·22-s − 0.208·23-s + 0.104·24-s + 0.122·25-s − 0.267·26-s + 0.564·27-s + 0.527·28-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 \( 1 + 0.510T + 3T^{2} \)
5 \( 1 - 2.36T + 5T^{2} \)
7 \( 1 - 2.78T + 7T^{2} \)
11 \( 1 - 0.898T + 11T^{2} \)
13 \( 1 - 1.36T + 13T^{2} \)
17 \( 1 + 4.57T + 17T^{2} \)
19 \( 1 - 3.53T + 19T^{2} \)
29 \( 1 + 9.50T + 29T^{2} \)
31 \( 1 + 7.83T + 31T^{2} \)
37 \( 1 + 3.08T + 37T^{2} \)
41 \( 1 - 6.50T + 41T^{2} \)
43 \( 1 + 2.06T + 43T^{2} \)
47 \( 1 - 2.55T + 47T^{2} \)
53 \( 1 + 0.0206T + 53T^{2} \)
59 \( 1 + 3.00T + 59T^{2} \)
61 \( 1 + 8.28T + 61T^{2} \)
67 \( 1 + 12.5T + 67T^{2} \)
71 \( 1 + 2.24T + 71T^{2} \)
73 \( 1 + 6.70T + 73T^{2} \)
79 \( 1 + 5.25T + 79T^{2} \)
83 \( 1 + 6.21T + 83T^{2} \)
89 \( 1 + 2.45T + 89T^{2} \)
97 \( 1 + 8.88T + 97T^{2} \)
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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

−7.71425597676455431397640566241, −7.16912493516013396197988930231, −6.13852134150440790244735710741, −5.74280455428181969552267539999, −5.10725736066889176512581664009, −4.06947295599066274383509409443, −2.96543162250558087383050718891, −1.97657387752740263827619880533, −1.48421042742276654901730865522, 0, 1.48421042742276654901730865522, 1.97657387752740263827619880533, 2.96543162250558087383050718891, 4.06947295599066274383509409443, 5.10725736066889176512581664009, 5.74280455428181969552267539999, 6.13852134150440790244735710741, 7.16912493516013396197988930231, 7.71425597676455431397640566241

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