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Properties

Label 2-26e2-13.9-c1-0-5
Degree $2$
Conductor $676$
Sign $0.859 - 0.511i$
Analytic cond. $5.39788$
Root an. cond. $2.32333$
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  + 2·5-s + (1 + 1.73i)7-s + (1.5 + 2.59i)9-s + (1 − 1.73i)11-s + (−3 − 5.19i)17-s + (3 + 5.19i)19-s + (−4 + 6.92i)23-s − 25-s + (−1 + 1.73i)29-s + 10·31-s + (2 + 3.46i)35-s + (3 − 5.19i)37-s + (3 − 5.19i)41-s + (−2 − 3.46i)43-s + (3 + 5.19i)45-s + ⋯
L(s)  = 1  + 0.894·5-s + (0.377 + 0.654i)7-s + (0.5 + 0.866i)9-s + (0.301 − 0.522i)11-s + (−0.727 − 1.26i)17-s + (0.688 + 1.19i)19-s + (−0.834 + 1.44i)23-s − 0.200·25-s + (−0.185 + 0.321i)29-s + 1.79·31-s + (0.338 + 0.585i)35-s + (0.493 − 0.854i)37-s + (0.468 − 0.811i)41-s + (−0.304 − 0.528i)43-s + (0.447 + 0.774i)45-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 676 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.859 - 0.511i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 676 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.859 - 0.511i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(676\)    =    \(2^{2} \cdot 13^{2}\)
Sign: $0.859 - 0.511i$
Analytic conductor: \(5.39788\)
Root analytic conductor: \(2.32333\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{676} (529, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 676,\ (\ :1/2),\ 0.859 - 0.511i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.80888 + 0.497142i\)
\(L(\frac12)\) \(\approx\) \(1.80888 + 0.497142i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
13 \( 1 \)
good3 \( 1 + (-1.5 - 2.59i)T^{2} \)
5 \( 1 - 2T + 5T^{2} \)
7 \( 1 + (-1 - 1.73i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1 + 1.73i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (3 + 5.19i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3 - 5.19i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (4 - 6.92i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1 - 1.73i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 10T + 31T^{2} \)
37 \( 1 + (-3 + 5.19i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-3 + 5.19i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (2 + 3.46i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 2T + 47T^{2} \)
53 \( 1 - 6T + 53T^{2} \)
59 \( 1 + (-5 - 8.66i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1 - 1.73i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (5 - 8.66i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (5 + 8.66i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 2T + 73T^{2} \)
79 \( 1 + 4T + 79T^{2} \)
83 \( 1 + 6T + 83T^{2} \)
89 \( 1 + (-3 + 5.19i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (1 + 1.73i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.39692297982278897478112204879, −9.752585407110636031026243756694, −8.943451830316586849713500732680, −7.967323186872632161068494912375, −7.10795890590299412586677499086, −5.80906191205362523421399047185, −5.38386760921177306659848086547, −4.11932333262077994974457587944, −2.60947219795475961566357644966, −1.60700338308445707531075240958, 1.17455097543999622918973221100, 2.48011681946557116919175285496, 4.05815452372245929095434204690, 4.73955054617102410807352979030, 6.28652908579985239071975431083, 6.58314087253901998840300588561, 7.82177861848072737912835206974, 8.777725825795076814792215086605, 9.790276531382174952774325682804, 10.15896970138293606442782531757

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