Properties

Label 2-13-1.1-c3-0-2
Degree $2$
Conductor $13$
Sign $-1$
Analytic cond. $0.767024$
Root an. cond. $0.875799$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Among degree 2 L-functions of (motivic) weight greater than 1, this is the one of positive rank with the smallest analytic conductor.

Dirichlet series

L(s)  = 1  − 5·2-s − 7·3-s + 17·4-s − 7·5-s + 35·6-s − 13·7-s − 45·8-s + 22·9-s + 35·10-s − 26·11-s − 119·12-s + 13·13-s + 65·14-s + 49·15-s + 89·16-s + 77·17-s − 110·18-s − 126·19-s − 119·20-s + 91·21-s + 130·22-s − 96·23-s + 315·24-s − 76·25-s − 65·26-s + 35·27-s − 221·28-s + ⋯
L(s)  = 1  − 1.76·2-s − 1.34·3-s + 17/8·4-s − 0.626·5-s + 2.38·6-s − 0.701·7-s − 1.98·8-s + 0.814·9-s + 1.10·10-s − 0.712·11-s − 2.86·12-s + 0.277·13-s + 1.24·14-s + 0.843·15-s + 1.39·16-s + 1.09·17-s − 1.44·18-s − 1.52·19-s − 1.33·20-s + 0.945·21-s + 1.25·22-s − 0.870·23-s + 2.67·24-s − 0.607·25-s − 0.490·26-s + 0.249·27-s − 1.49·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(13\)
Sign: $-1$
Analytic conductor: \(0.767024\)
Root analytic conductor: \(0.875799\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: $\chi_{13} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 13,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 - p T \)
good2 \( 1 + 5 T + p^{3} T^{2} \)
3 \( 1 + 7 T + p^{3} T^{2} \)
5 \( 1 + 7 T + p^{3} T^{2} \)
7 \( 1 + 13 T + p^{3} T^{2} \)
11 \( 1 + 26 T + p^{3} T^{2} \)
17 \( 1 - 77 T + p^{3} T^{2} \)
19 \( 1 + 126 T + p^{3} T^{2} \)
23 \( 1 + 96 T + p^{3} T^{2} \)
29 \( 1 + 82 T + p^{3} T^{2} \)
31 \( 1 - 196 T + p^{3} T^{2} \)
37 \( 1 + 131 T + p^{3} T^{2} \)
41 \( 1 - 336 T + p^{3} T^{2} \)
43 \( 1 + 201 T + p^{3} T^{2} \)
47 \( 1 + 105 T + p^{3} T^{2} \)
53 \( 1 + 432 T + p^{3} T^{2} \)
59 \( 1 + 294 T + p^{3} T^{2} \)
61 \( 1 + 56 T + p^{3} T^{2} \)
67 \( 1 - 478 T + p^{3} T^{2} \)
71 \( 1 - 9 T + p^{3} T^{2} \)
73 \( 1 - 98 T + p^{3} T^{2} \)
79 \( 1 - 1304 T + p^{3} T^{2} \)
83 \( 1 + 308 T + p^{3} T^{2} \)
89 \( 1 + 1190 T + p^{3} T^{2} \)
97 \( 1 - 70 T + p^{3} T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−18.64002945313011709440369128241, −17.47231804263601398254082417282, −16.49731910284015820282370154310, −15.66451581817735663308079993374, −12.33586951176125151168612275176, −11.07933729517739416617187501775, −10.00504625289844338036427985935, −8.015540090380562026102644883058, −6.30516929548526949298373195082, 0, 6.30516929548526949298373195082, 8.015540090380562026102644883058, 10.00504625289844338036427985935, 11.07933729517739416617187501775, 12.33586951176125151168612275176, 15.66451581817735663308079993374, 16.49731910284015820282370154310, 17.47231804263601398254082417282, 18.64002945313011709440369128241

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