Properties

Degree $2$
Conductor $26$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 3·3-s + 4-s − 5-s − 3·6-s + 7-s + 8-s + 6·9-s − 10-s − 2·11-s − 3·12-s − 13-s + 14-s + 3·15-s + 16-s − 3·17-s + 6·18-s + 6·19-s − 20-s − 3·21-s − 2·22-s − 4·23-s − 3·24-s − 4·25-s − 26-s − 9·27-s + 28-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.73·3-s + 1/2·4-s − 0.447·5-s − 1.22·6-s + 0.377·7-s + 0.353·8-s + 2·9-s − 0.316·10-s − 0.603·11-s − 0.866·12-s − 0.277·13-s + 0.267·14-s + 0.774·15-s + 1/4·16-s − 0.727·17-s + 1.41·18-s + 1.37·19-s − 0.223·20-s − 0.654·21-s − 0.426·22-s − 0.834·23-s − 0.612·24-s − 4/5·25-s − 0.196·26-s − 1.73·27-s + 0.188·28-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: $1$
Motivic weight: \(1\)
Character: $\chi_{26} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 26,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6209653495\)
\(L(\frac12)\) \(\approx\) \(0.6209653495\)
\(L(\frac{3}{2})\) 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 \( 1 + p T + p T^{2} \)
5 \( 1 + T + p T^{2} \)
7 \( 1 - T + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 3 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 5 T + p T^{2} \)
47 \( 1 - 13 T + p T^{2} \)
53 \( 1 - 12 T + p T^{2} \)
59 \( 1 + 10 T + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 + 5 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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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

−18.38839909652752, −17.46439533057334, −16.21055196648710, −15.44251689428644, −13.59228638804353, −12.14995362405969, −11.49712673239204, −10.28025762974353, −7.459855606675890, −5.889084509159496, −4.611534534911821, 4.611534534911821, 5.889084509159496, 7.459855606675890, 10.28025762974353, 11.49712673239204, 12.14995362405969, 13.59228638804353, 15.44251689428644, 16.21055196648710, 17.46439533057334, 18.38839909652752

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