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Properties

Label 2-26e2-52.51-c0-0-1
Degree $2$
Conductor $676$
Sign $0.832 - 0.554i$
Analytic cond. $0.337367$
Root an. cond. $0.580833$
Motivic weight $0$
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 − 4-s − i·5-s − i·8-s + 9-s + 10-s + 16-s + 17-s + i·18-s + i·20-s − 29-s + i·32-s + i·34-s − 36-s + i·37-s + ⋯
L(s)  = 1  + i·2-s − 4-s − i·5-s − i·8-s + 9-s + 10-s + 16-s + 17-s + i·18-s + i·20-s − 29-s + i·32-s + i·34-s − 36-s + i·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 676 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.832 - 0.554i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 676 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.832 - 0.554i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(676\)    =    \(2^{2} \cdot 13^{2}\)
Sign: $0.832 - 0.554i$
Analytic conductor: \(0.337367\)
Root analytic conductor: \(0.580833\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{676} (675, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 676,\ (\ :0),\ 0.832 - 0.554i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9125552380\)
\(L(\frac12)\) \(\approx\) \(0.9125552380\)
\(L(1)\) 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 \( 1 \)
good3 \( 1 - T^{2} \)
5 \( 1 + iT - T^{2} \)
7 \( 1 + T^{2} \)
11 \( 1 + T^{2} \)
17 \( 1 - T + T^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + T + T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 - iT - T^{2} \)
41 \( 1 + iT - T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + T + T^{2} \)
59 \( 1 + T^{2} \)
61 \( 1 + T + T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 - iT - T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + 2iT - T^{2} \)
97 \( 1 - 2iT - T^{2} \)
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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

−10.46870693552330568644087118685, −9.658202526604749882294129989996, −9.001450219269808719004164924414, −8.049231277810955034779837119765, −7.38639700795928015494000202826, −6.36258253437526093266638341539, −5.31426646929877387509070449192, −4.62324686760569529842796746024, −3.59096776911207756611741946119, −1.32004426888021298833398603327, 1.62826427500627072966391379375, 2.95028313491531740824300178130, 3.81464171922524960755544021108, 4.91554474485232945717649467416, 6.10443864946433196507472156121, 7.26072005063231472595606729850, 8.007991091956880870663342401106, 9.311721818619462828716432241233, 9.906183274459220138093506176539, 10.69108350503234792112174303004

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