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Properties

Label 2-776-776.339-c0-0-0
Degree $2$
Conductor $776$
Sign $0.871 + 0.491i$
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  + (0.991 − 0.130i)2-s + (−0.207 − 0.158i)3-s + (0.965 − 0.258i)4-s + (−0.226 − 0.130i)6-s + (0.923 − 0.382i)8-s + (−0.241 − 0.900i)9-s + (−0.0675 + 0.513i)11-s + (−0.241 − 0.0999i)12-s + (0.866 − 0.5i)16-s + (−0.389 + 0.0255i)17-s + (−0.356 − 0.860i)18-s + (−0.923 + 0.617i)19-s + 0.517i·22-s + (−0.252 − 0.0675i)24-s + (0.608 + 0.793i)25-s + ⋯
L(s)  = 1  + (0.991 − 0.130i)2-s + (−0.207 − 0.158i)3-s + (0.965 − 0.258i)4-s + (−0.226 − 0.130i)6-s + (0.923 − 0.382i)8-s + (−0.241 − 0.900i)9-s + (−0.0675 + 0.513i)11-s + (−0.241 − 0.0999i)12-s + (0.866 − 0.5i)16-s + (−0.389 + 0.0255i)17-s + (−0.356 − 0.860i)18-s + (−0.923 + 0.617i)19-s + 0.517i·22-s + (−0.252 − 0.0675i)24-s + (0.608 + 0.793i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 776 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.871 + 0.491i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 776 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.871 + 0.491i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(776\)    =    \(2^{3} \cdot 97\)
Sign: $0.871 + 0.491i$
Analytic conductor: \(0.387274\)
Root analytic conductor: \(0.622313\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{776} (339, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 776,\ (\ :0),\ 0.871 + 0.491i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.604764322\)
\(L(\frac12)\) \(\approx\) \(1.604764322\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.991 + 0.130i)T \)
97 \( 1 + (-0.793 - 0.608i)T \)
good3 \( 1 + (0.207 + 0.158i)T + (0.258 + 0.965i)T^{2} \)
5 \( 1 + (-0.608 - 0.793i)T^{2} \)
7 \( 1 + (0.991 + 0.130i)T^{2} \)
11 \( 1 + (0.0675 - 0.513i)T + (-0.965 - 0.258i)T^{2} \)
13 \( 1 + (0.608 + 0.793i)T^{2} \)
17 \( 1 + (0.389 - 0.0255i)T + (0.991 - 0.130i)T^{2} \)
19 \( 1 + (0.923 - 0.617i)T + (0.382 - 0.923i)T^{2} \)
23 \( 1 + (-0.130 - 0.991i)T^{2} \)
29 \( 1 + (-0.793 + 0.608i)T^{2} \)
31 \( 1 + (0.258 + 0.965i)T^{2} \)
37 \( 1 + (0.130 - 0.991i)T^{2} \)
41 \( 1 + (0.534 + 1.57i)T + (-0.793 + 0.608i)T^{2} \)
43 \( 1 + (0.410 - 1.53i)T + (-0.866 - 0.5i)T^{2} \)
47 \( 1 - iT^{2} \)
53 \( 1 + (0.965 - 0.258i)T^{2} \)
59 \( 1 + (1.47 - 1.29i)T + (0.130 - 0.991i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.357 + 0.534i)T + (-0.382 + 0.923i)T^{2} \)
71 \( 1 + (0.793 + 0.608i)T^{2} \)
73 \( 1 + (-1.36 - 0.366i)T + (0.866 + 0.5i)T^{2} \)
79 \( 1 + (0.707 - 0.707i)T^{2} \)
83 \( 1 + (0.0862 + 1.31i)T + (-0.991 + 0.130i)T^{2} \)
89 \( 1 + (1.12 - 0.465i)T + (0.707 - 0.707i)T^{2} \)
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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

−10.69041569980470957244158948720, −9.770060083602051347067589064502, −8.768553755967614813381446301902, −7.58677397816283211333748080517, −6.69373258956802526879205841463, −6.05705811292177314854903830722, −5.02270454953911931063730809798, −4.04876080781768195012888530978, −3.05369187687545123138648723073, −1.69374815170021849135477260065, 2.10276769839194210628504296801, 3.15993318808194880853596648648, 4.45173639333454768307881043434, 5.06895571403578919611452382334, 6.12134777878106563035381809370, 6.83737955274419454452791761978, 7.973514560735670048971840698556, 8.629098223951214906693727905787, 10.03403188623582617867909385433, 10.91779817136213741935096723608

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