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 − 0.666·3-s + 4-s + 0.513·5-s + 0.666·6-s + 2.90·7-s − 8-s − 2.55·9-s − 0.513·10-s + 0.858·11-s − 0.666·12-s − 5.16·13-s − 2.90·14-s − 0.342·15-s + 16-s + 4.36·17-s + 2.55·18-s − 1.53·19-s + 0.513·20-s − 1.93·21-s − 0.858·22-s − 23-s + 0.666·24-s − 4.73·25-s + 5.16·26-s + 3.70·27-s + 2.90·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.384·3-s + 0.5·4-s + 0.229·5-s + 0.271·6-s + 1.09·7-s − 0.353·8-s − 0.852·9-s − 0.162·10-s + 0.258·11-s − 0.192·12-s − 1.43·13-s − 0.775·14-s − 0.0883·15-s + 0.250·16-s + 1.05·17-s + 0.602·18-s − 0.353·19-s + 0.114·20-s − 0.421·21-s − 0.183·22-s − 0.208·23-s + 0.135·24-s − 0.947·25-s + 1.01·26-s + 0.712·27-s + 0.548·28-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 \( 1 + 0.666T + 3T^{2} \)
5 \( 1 - 0.513T + 5T^{2} \)
7 \( 1 - 2.90T + 7T^{2} \)
11 \( 1 - 0.858T + 11T^{2} \)
13 \( 1 + 5.16T + 13T^{2} \)
17 \( 1 - 4.36T + 17T^{2} \)
19 \( 1 + 1.53T + 19T^{2} \)
29 \( 1 + 3.13T + 29T^{2} \)
31 \( 1 - 9.58T + 31T^{2} \)
37 \( 1 + 7.80T + 37T^{2} \)
41 \( 1 - 4.19T + 41T^{2} \)
43 \( 1 - 11.0T + 43T^{2} \)
47 \( 1 + 1.61T + 47T^{2} \)
53 \( 1 + 6.05T + 53T^{2} \)
59 \( 1 - 3.21T + 59T^{2} \)
61 \( 1 + 6.32T + 61T^{2} \)
67 \( 1 + 3.00T + 67T^{2} \)
71 \( 1 - 2.11T + 71T^{2} \)
73 \( 1 - 12.7T + 73T^{2} \)
79 \( 1 + 2.80T + 79T^{2} \)
83 \( 1 + 16.3T + 83T^{2} \)
89 \( 1 + 10.3T + 89T^{2} \)
97 \( 1 + 13.5T + 97T^{2} \)
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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

−7.85911210833871846361089662055, −7.20832148074088115866585424337, −6.28460021773355756224146114606, −5.60532910709166266485315332877, −5.02170511145157460407069769366, −4.15267705898469082568963335144, −2.92677988388254805051053829658, −2.19607620679249243580155242508, −1.23180008523943881369870375559, 0, 1.23180008523943881369870375559, 2.19607620679249243580155242508, 2.92677988388254805051053829658, 4.15267705898469082568963335144, 5.02170511145157460407069769366, 5.60532910709166266485315332877, 6.28460021773355756224146114606, 7.20832148074088115866585424337, 7.85911210833871846361089662055

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