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 − 1.05·3-s + 4-s − 2.13·5-s + 1.05·6-s + 3.42·7-s − 8-s − 1.88·9-s + 2.13·10-s − 2.88·11-s − 1.05·12-s + 1.88·13-s − 3.42·14-s + 2.26·15-s + 16-s + 4.09·17-s + 1.88·18-s − 3.98·19-s − 2.13·20-s − 3.62·21-s + 2.88·22-s + 23-s + 1.05·24-s − 0.428·25-s − 1.88·26-s + 5.16·27-s + 3.42·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.610·3-s + 0.5·4-s − 0.956·5-s + 0.431·6-s + 1.29·7-s − 0.353·8-s − 0.627·9-s + 0.676·10-s − 0.871·11-s − 0.305·12-s + 0.522·13-s − 0.915·14-s + 0.583·15-s + 0.250·16-s + 0.993·17-s + 0.443·18-s − 0.915·19-s − 0.478·20-s − 0.790·21-s + 0.615·22-s + 0.208·23-s + 0.215·24-s − 0.0856·25-s − 0.369·26-s + 0.993·27-s + 0.647·28-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 \( 1 + 1.05T + 3T^{2} \)
5 \( 1 + 2.13T + 5T^{2} \)
7 \( 1 - 3.42T + 7T^{2} \)
11 \( 1 + 2.88T + 11T^{2} \)
13 \( 1 - 1.88T + 13T^{2} \)
17 \( 1 - 4.09T + 17T^{2} \)
19 \( 1 + 3.98T + 19T^{2} \)
29 \( 1 - 7.47T + 29T^{2} \)
31 \( 1 + 8.99T + 31T^{2} \)
37 \( 1 + 5.04T + 37T^{2} \)
41 \( 1 - 8.18T + 41T^{2} \)
43 \( 1 - 0.248T + 43T^{2} \)
47 \( 1 + 6.96T + 47T^{2} \)
53 \( 1 - 6.38T + 53T^{2} \)
59 \( 1 - 11.2T + 59T^{2} \)
61 \( 1 + 1.76T + 61T^{2} \)
67 \( 1 + 15.5T + 67T^{2} \)
71 \( 1 - 9.84T + 71T^{2} \)
73 \( 1 - 2.29T + 73T^{2} \)
79 \( 1 - 0.874T + 79T^{2} \)
83 \( 1 + 9.68T + 83T^{2} \)
89 \( 1 + 0.610T + 89T^{2} \)
97 \( 1 - 5.60T + 97T^{2} \)
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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

−7.77023402022725498659202808100, −7.32596514780035934138599162661, −6.32311908294611297497022185281, −5.54119491776899447979682477226, −4.99547467928131889233615760697, −4.08744683460138472351676023144, −3.15802985619715790107922631478, −2.14885845714706907506718338076, −1.04838368086083808633468380519, 0, 1.04838368086083808633468380519, 2.14885845714706907506718338076, 3.15802985619715790107922631478, 4.08744683460138472351676023144, 4.99547467928131889233615760697, 5.54119491776899447979682477226, 6.32311908294611297497022185281, 7.32596514780035934138599162661, 7.77023402022725498659202808100

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