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Properties

Label 2-776-776.11-c0-0-0
Degree $2$
Conductor $776$
Sign $-0.881 + 0.471i$
Analytic cond. $0.387274$
Root an. cond. $0.622313$
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  + (−0.130 − 0.991i)2-s + (1.20 − 1.57i)3-s + (−0.965 + 0.258i)4-s + (−1.71 − 0.991i)6-s + (0.382 + 0.923i)8-s + (−0.758 − 2.83i)9-s + (0.513 + 0.0675i)11-s + (−0.758 + 1.83i)12-s + (0.866 − 0.5i)16-s + (−0.732 + 0.835i)17-s + (−2.70 + 1.12i)18-s + (−0.382 + 1.92i)19-s − 0.517i·22-s + (1.91 + 0.513i)24-s + (0.793 − 0.608i)25-s + ⋯
L(s)  = 1  + (−0.130 − 0.991i)2-s + (1.20 − 1.57i)3-s + (−0.965 + 0.258i)4-s + (−1.71 − 0.991i)6-s + (0.382 + 0.923i)8-s + (−0.758 − 2.83i)9-s + (0.513 + 0.0675i)11-s + (−0.758 + 1.83i)12-s + (0.866 − 0.5i)16-s + (−0.732 + 0.835i)17-s + (−2.70 + 1.12i)18-s + (−0.382 + 1.92i)19-s − 0.517i·22-s + (1.91 + 0.513i)24-s + (0.793 − 0.608i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 776 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.881 + 0.471i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 776 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.881 + 0.471i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(776\)    =    \(2^{3} \cdot 97\)
Sign: $-0.881 + 0.471i$
Analytic conductor: \(0.387274\)
Root analytic conductor: \(0.622313\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{776} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 776,\ (\ :0),\ -0.881 + 0.471i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.179995148\)
\(L(\frac12)\) \(\approx\) \(1.179995148\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.130 + 0.991i)T \)
97 \( 1 + (0.608 - 0.793i)T \)
good3 \( 1 + (-1.20 + 1.57i)T + (-0.258 - 0.965i)T^{2} \)
5 \( 1 + (-0.793 + 0.608i)T^{2} \)
7 \( 1 + (-0.130 + 0.991i)T^{2} \)
11 \( 1 + (-0.513 - 0.0675i)T + (0.965 + 0.258i)T^{2} \)
13 \( 1 + (0.793 - 0.608i)T^{2} \)
17 \( 1 + (0.732 - 0.835i)T + (-0.130 - 0.991i)T^{2} \)
19 \( 1 + (0.382 - 1.92i)T + (-0.923 - 0.382i)T^{2} \)
23 \( 1 + (-0.991 + 0.130i)T^{2} \)
29 \( 1 + (0.608 + 0.793i)T^{2} \)
31 \( 1 + (-0.258 - 0.965i)T^{2} \)
37 \( 1 + (0.991 + 0.130i)T^{2} \)
41 \( 1 + (-0.349 - 0.172i)T + (0.608 + 0.793i)T^{2} \)
43 \( 1 + (0.315 - 1.17i)T + (-0.866 - 0.5i)T^{2} \)
47 \( 1 - iT^{2} \)
53 \( 1 + (-0.965 + 0.258i)T^{2} \)
59 \( 1 + (1.65 + 0.108i)T + (0.991 + 0.130i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (-1.75 - 0.349i)T + (0.923 + 0.382i)T^{2} \)
71 \( 1 + (-0.608 + 0.793i)T^{2} \)
73 \( 1 + (-1.36 - 0.366i)T + (0.866 + 0.5i)T^{2} \)
79 \( 1 + (-0.707 + 0.707i)T^{2} \)
83 \( 1 + (0.0983 + 0.0862i)T + (0.130 + 0.991i)T^{2} \)
89 \( 1 + (0.607 + 1.46i)T + (-0.707 + 0.707i)T^{2} \)
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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

−10.01643685946589333858508176161, −9.190042847531987835781929324337, −8.319950840338825071291573163204, −8.041392345462483885197660235323, −6.81701741042954916852265537575, −5.98205176005426048465331721770, −4.14937963653649571973124127663, −3.30692934130950904859762348730, −2.18840753687828050195450009974, −1.37961915618935007594675964211, 2.61493525083454288700410068608, 3.75071226916414568559842951635, 4.66872388988798596784589912122, 5.18500761716613841734249261944, 6.68623520737840671303329939897, 7.58743321629371212127117392055, 8.615530517640628668987335558439, 9.138871098517266359938900036437, 9.516330563575990139410942409596, 10.64170815709288340444248938176

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