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Properties

Label 2-26-13.12-c1-0-1
Degree $2$
Conductor $26$
Sign $0.832 + 0.554i$
Analytic cond. $0.207611$
Root an. cond. $0.455643$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − i·2-s − 3-s − 4-s + 3i·5-s + i·6-s − 3i·7-s + i·8-s − 2·9-s + 3·10-s + 12-s + (2 − 3i)13-s − 3·14-s − 3i·15-s + 16-s + 3·17-s + 2i·18-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.577·3-s − 0.5·4-s + 1.34i·5-s + 0.408i·6-s − 1.13i·7-s + 0.353i·8-s − 0.666·9-s + 0.948·10-s + 0.288·12-s + (0.554 − 0.832i)13-s − 0.801·14-s − 0.774i·15-s + 0.250·16-s + 0.727·17-s + 0.471i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 26 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.832 + 0.554i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.832 + 0.554i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(26\)    =    \(2 \cdot 13\)
Sign: $0.832 + 0.554i$
Analytic conductor: \(0.207611\)
Root analytic conductor: \(0.455643\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{26} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 26,\ (\ :1/2),\ 0.832 + 0.554i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.538704 - 0.163106i\)
\(L(\frac12)\) \(\approx\) \(0.538704 - 0.163106i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
13 \( 1 + (-2 + 3i)T \)
good3 \( 1 + T + 3T^{2} \)
5 \( 1 - 3iT - 5T^{2} \)
7 \( 1 + 3iT - 7T^{2} \)
11 \( 1 - 11T^{2} \)
17 \( 1 - 3T + 17T^{2} \)
19 \( 1 - 6iT - 19T^{2} \)
23 \( 1 + 6T + 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 + 3iT - 37T^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 + T + 43T^{2} \)
47 \( 1 + 3iT - 47T^{2} \)
53 \( 1 + 6T + 53T^{2} \)
59 \( 1 - 6iT - 59T^{2} \)
61 \( 1 + 8T + 61T^{2} \)
67 \( 1 - 12iT - 67T^{2} \)
71 \( 1 + 15iT - 71T^{2} \)
73 \( 1 + 6iT - 73T^{2} \)
79 \( 1 - 10T + 79T^{2} \)
83 \( 1 + 6iT - 83T^{2} \)
89 \( 1 - 6iT - 89T^{2} \)
97 \( 1 - 12iT - 97T^{2} \)
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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

−17.69518615041136556702863141543, −16.49082166238566739681079280254, −14.62874621167701846399328933010, −13.77322524224678752979164692856, −12.06840841229674551327782820165, −10.78604730673694910604982598109, −10.24097835334179608955390309155, −7.81580411016520448253038462683, −6.01632195233972618102175746207, −3.49748557902694879901443144890, 4.94931671065091800429494720066, 6.10272337101188958807676622402, 8.380034989348270758343219773095, 9.273169857449595050678731052098, 11.58320108030543861280167783156, 12.56955955660686972306929715682, 14.00419847866366404346413565835, 15.59041563034695015036545603647, 16.44997075291564349503830075472, 17.36882624358461094367934412757

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