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Properties

Label 2-776-776.323-c0-0-0
Degree $2$
Conductor $776$
Sign $0.904 + 0.426i$
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  + (−0.793 − 0.608i)2-s + (1.20 − 0.158i)3-s + (0.258 + 0.965i)4-s + (−1.05 − 0.608i)6-s + (0.382 − 0.923i)8-s + (0.465 − 0.124i)9-s + (1.17 + 1.53i)11-s + (0.465 + 1.12i)12-s + (−0.866 + 0.499i)16-s + (−0.357 + 1.05i)17-s + (−0.445 − 0.184i)18-s + (−0.382 − 1.92i)19-s − 1.93i·22-s + (0.315 − 1.17i)24-s + (0.130 − 0.991i)25-s + ⋯
L(s)  = 1  + (−0.793 − 0.608i)2-s + (1.20 − 0.158i)3-s + (0.258 + 0.965i)4-s + (−1.05 − 0.608i)6-s + (0.382 − 0.923i)8-s + (0.465 − 0.124i)9-s + (1.17 + 1.53i)11-s + (0.465 + 1.12i)12-s + (−0.866 + 0.499i)16-s + (−0.357 + 1.05i)17-s + (−0.445 − 0.184i)18-s + (−0.382 − 1.92i)19-s − 1.93i·22-s + (0.315 − 1.17i)24-s + (0.130 − 0.991i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 776 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.904 + 0.426i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 776 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.904 + 0.426i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(776\)    =    \(2^{3} \cdot 97\)
Sign: $0.904 + 0.426i$
Analytic conductor: \(0.387274\)
Root analytic conductor: \(0.622313\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{776} (323, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 776,\ (\ :0),\ 0.904 + 0.426i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9935278803\)
\(L(\frac12)\) \(\approx\) \(0.9935278803\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.793 + 0.608i)T \)
97 \( 1 + (-0.991 + 0.130i)T \)
good3 \( 1 + (-1.20 + 0.158i)T + (0.965 - 0.258i)T^{2} \)
5 \( 1 + (-0.130 + 0.991i)T^{2} \)
7 \( 1 + (-0.793 + 0.608i)T^{2} \)
11 \( 1 + (-1.17 - 1.53i)T + (-0.258 + 0.965i)T^{2} \)
13 \( 1 + (0.130 - 0.991i)T^{2} \)
17 \( 1 + (0.357 - 1.05i)T + (-0.793 - 0.608i)T^{2} \)
19 \( 1 + (0.382 + 1.92i)T + (-0.923 + 0.382i)T^{2} \)
23 \( 1 + (0.608 - 0.793i)T^{2} \)
29 \( 1 + (-0.991 - 0.130i)T^{2} \)
31 \( 1 + (0.965 - 0.258i)T^{2} \)
37 \( 1 + (-0.608 - 0.793i)T^{2} \)
41 \( 1 + (0.0255 - 0.389i)T + (-0.991 - 0.130i)T^{2} \)
43 \( 1 + (1.91 + 0.513i)T + (0.866 + 0.5i)T^{2} \)
47 \( 1 + iT^{2} \)
53 \( 1 + (0.258 + 0.965i)T^{2} \)
59 \( 1 + (-0.735 + 1.49i)T + (-0.608 - 0.793i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.128 - 0.0255i)T + (0.923 - 0.382i)T^{2} \)
71 \( 1 + (0.991 - 0.130i)T^{2} \)
73 \( 1 + (0.366 - 1.36i)T + (-0.866 - 0.5i)T^{2} \)
79 \( 1 + (-0.707 - 0.707i)T^{2} \)
83 \( 1 + (1.69 + 0.576i)T + (0.793 + 0.608i)T^{2} \)
89 \( 1 + (0.0999 - 0.241i)T + (-0.707 - 0.707i)T^{2} \)
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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

−10.16813125281011969716454701592, −9.514935529477012364449673619533, −8.746981978920995056440522578062, −8.291162028065923632975397871208, −7.10963675646665646529005097482, −6.66882019429154670967168502148, −4.58825887733195187699118566884, −3.75828786621767776943095163086, −2.54121745258923744840732474087, −1.76532367359657227337553504339, 1.53226258952428566210194819814, 3.00024912088580320235515188920, 3.98266152628328946630297041960, 5.52915070817034283960867084422, 6.33807708352971229351812572914, 7.37354261618838088376523746638, 8.266853199295251822177847059435, 8.787713465133200976925801812314, 9.389131468626207608230092584885, 10.23239923930872457775013744262

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