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Properties

Label 2-28e2-112.61-c0-0-0
Degree $2$
Conductor $784$
Sign $0.557 - 0.830i$
Analytic cond. $0.391266$
Root an. cond. $0.625513$
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  + (0.866 + 0.5i)2-s + (0.499 + 0.866i)4-s + 0.999i·8-s + (−0.866 − 0.5i)9-s + (1.36 + 0.366i)11-s + (−0.5 + 0.866i)16-s + (−0.499 − 0.866i)18-s + (0.999 + i)22-s + (−0.866 + 0.5i)25-s + (−1 − i)29-s + (−0.866 + 0.499i)32-s − 0.999i·36-s + (−0.366 − 1.36i)37-s + (1 − i)43-s + (0.366 + 1.36i)44-s + ⋯
L(s)  = 1  + (0.866 + 0.5i)2-s + (0.499 + 0.866i)4-s + 0.999i·8-s + (−0.866 − 0.5i)9-s + (1.36 + 0.366i)11-s + (−0.5 + 0.866i)16-s + (−0.499 − 0.866i)18-s + (0.999 + i)22-s + (−0.866 + 0.5i)25-s + (−1 − i)29-s + (−0.866 + 0.499i)32-s − 0.999i·36-s + (−0.366 − 1.36i)37-s + (1 − i)43-s + (0.366 + 1.36i)44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.557 - 0.830i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.557 - 0.830i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(784\)    =    \(2^{4} \cdot 7^{2}\)
Sign: $0.557 - 0.830i$
Analytic conductor: \(0.391266\)
Root analytic conductor: \(0.625513\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{784} (509, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 784,\ (\ :0),\ 0.557 - 0.830i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.536176895\)
\(L(\frac12)\) \(\approx\) \(1.536176895\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 - 0.5i)T \)
7 \( 1 \)
good3 \( 1 + (0.866 + 0.5i)T^{2} \)
5 \( 1 + (0.866 - 0.5i)T^{2} \)
11 \( 1 + (-1.36 - 0.366i)T + (0.866 + 0.5i)T^{2} \)
13 \( 1 - iT^{2} \)
17 \( 1 + (0.5 - 0.866i)T^{2} \)
19 \( 1 + (-0.866 + 0.5i)T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 + (1 + i)T + iT^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 + (0.366 + 1.36i)T + (-0.866 + 0.5i)T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + (-1 + i)T - iT^{2} \)
47 \( 1 + (0.5 + 0.866i)T^{2} \)
53 \( 1 + (1.36 + 0.366i)T + (0.866 + 0.5i)T^{2} \)
59 \( 1 + (-0.866 - 0.5i)T^{2} \)
61 \( 1 + (-0.866 + 0.5i)T^{2} \)
67 \( 1 + (-0.366 + 1.36i)T + (-0.866 - 0.5i)T^{2} \)
71 \( 1 - 2iT - T^{2} \)
73 \( 1 + (-0.5 + 0.866i)T^{2} \)
79 \( 1 + (-0.5 - 0.866i)T^{2} \)
83 \( 1 - iT^{2} \)
89 \( 1 + (-0.5 - 0.866i)T^{2} \)
97 \( 1 - T^{2} \)
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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

−11.00009891457153396929087863757, −9.535339031556188954826903858473, −8.919139121751317029112925721045, −7.87746619065921854982091127240, −7.00758722936729573820396884627, −6.13787462419976574452945704401, −5.47374405527235832983959515636, −4.15669178079296338305898078564, −3.51759229120207905617348263878, −2.10891053860849463854351551763, 1.59605211062754031598975477817, 2.92804047934316844424894646980, 3.86972376788839363138347039731, 4.90307135134489765029988769779, 5.90089983518071369568057180241, 6.54319224680316589806691845089, 7.71837527427162723931343794568, 8.854688866537050987776744373975, 9.606412748917052530746680776265, 10.64556640575123947814497873046

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