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Properties

Label 2-777-777.158-c0-0-1
Degree $2$
Conductor $777$
Sign $-0.425 + 0.904i$
Analytic cond. $0.387773$
Root an. cond. $0.622714$
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·3-s + 4-s + (−0.866 − 0.5i)5-s − 7-s − 9-s + (−0.866 − 0.5i)11-s − i·12-s + (−0.5 − 0.866i)13-s + (−0.5 + 0.866i)15-s + 16-s + (0.866 − 0.5i)17-s + (0.5 − 0.866i)19-s + (−0.866 − 0.5i)20-s + i·21-s + (0.866 − 0.5i)23-s + ⋯
L(s)  = 1  − i·3-s + 4-s + (−0.866 − 0.5i)5-s − 7-s − 9-s + (−0.866 − 0.5i)11-s − i·12-s + (−0.5 − 0.866i)13-s + (−0.5 + 0.866i)15-s + 16-s + (0.866 − 0.5i)17-s + (0.5 − 0.866i)19-s + (−0.866 − 0.5i)20-s + i·21-s + (0.866 − 0.5i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 777 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.425 + 0.904i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 777 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.425 + 0.904i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(777\)    =    \(3 \cdot 7 \cdot 37\)
Sign: $-0.425 + 0.904i$
Analytic conductor: \(0.387773\)
Root analytic conductor: \(0.622714\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{777} (158, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 777,\ (\ :0),\ -0.425 + 0.904i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8346841669\)
\(L(\frac12)\) \(\approx\) \(0.8346841669\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + iT \)
7 \( 1 + T \)
37 \( 1 - T \)
good2 \( 1 - T^{2} \)
5 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
13 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
17 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
19 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
29 \( 1 - 2iT - T^{2} \)
31 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
41 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
53 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
59 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
73 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
89 \( 1 - T^{2} \)
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

−10.56271922299431679950849230552, −9.238139817809808856370757082487, −8.294030099393861070990575819998, −7.39576925460443540002846162162, −7.11253197288260324267231739481, −5.90286912762541436931993687835, −5.13902203069845126958730333956, −3.16227268143647418575973658545, −2.81647442383636142731950778919, −0.853010261749059601201612908478, 2.48372778569171629528473093584, 3.36193549677754432965488952724, 4.18734826718923280178088039377, 5.58777094551785615226960086802, 6.34929452452906977590734573935, 7.55885795760214216649914123568, 7.87132700261184303152701901046, 9.587438680777873206165674932996, 9.812333486022521307558257756203, 10.83567710279079877537553405202

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