Properties

Label 2-26-1.1-c1-0-0
Degree $2$
Conductor $26$
Sign $1$
Analytic cond. $0.207611$
Root an. cond. $0.455643$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 3-s + 4-s − 3·5-s − 6-s − 7-s − 8-s − 2·9-s + 3·10-s + 6·11-s + 12-s + 13-s + 14-s − 3·15-s + 16-s − 3·17-s + 2·18-s + 2·19-s − 3·20-s − 21-s − 6·22-s − 24-s + 4·25-s − 26-s − 5·27-s − 28-s + 6·29-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s − 1.34·5-s − 0.408·6-s − 0.377·7-s − 0.353·8-s − 2/3·9-s + 0.948·10-s + 1.80·11-s + 0.288·12-s + 0.277·13-s + 0.267·14-s − 0.774·15-s + 1/4·16-s − 0.727·17-s + 0.471·18-s + 0.458·19-s − 0.670·20-s − 0.218·21-s − 1.27·22-s − 0.204·24-s + 4/5·25-s − 0.196·26-s − 0.962·27-s − 0.188·28-s + 1.11·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 26 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(26\)    =    \(2 \cdot 13\)
Sign: $1$
Analytic conductor: \(0.207611\)
Root analytic conductor: \(0.455643\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 26,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5155766512\)
\(L(\frac12)\) \(\approx\) \(0.5155766512\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
13 \( 1 - T \)
good3 \( 1 - T + p T^{2} \)
5 \( 1 + 3 T + p T^{2} \)
7 \( 1 + T + p T^{2} \)
11 \( 1 - 6 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 7 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + T + p T^{2} \)
47 \( 1 - 3 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 - 14 T + p T^{2} \)
71 \( 1 + 3 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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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

−17.51877200404443387419304773878, −16.32893427071430805415502474938, −15.26212233610873072978362349109, −14.10812305484994740358213704547, −12.10810868275673738373780840946, −11.21486999262721520146961750481, −9.252086119982469784390910809299, −8.289125051595985253760732717123, −6.76423132327857768444846951818, −3.64028761626013442697591583469, 3.64028761626013442697591583469, 6.76423132327857768444846951818, 8.289125051595985253760732717123, 9.252086119982469784390910809299, 11.21486999262721520146961750481, 12.10810868275673738373780840946, 14.10812305484994740358213704547, 15.26212233610873072978362349109, 16.32893427071430805415502474938, 17.51877200404443387419304773878

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