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Properties

Label 2-776-776.243-c0-0-0
Degree $2$
Conductor $776$
Sign $0.544 + 0.838i$
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 + (−0.128 − 1.95i)17-s + (0.356 + 0.860i)18-s + (0.923 + 1.38i)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 + (−0.128 − 1.95i)17-s + (0.356 + 0.860i)18-s + (0.923 + 1.38i)19-s + 0.517i·22-s + (0.252 + 0.0675i)24-s + (−0.608 − 0.793i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 776 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.544 + 0.838i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 776 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.544 + 0.838i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(776\)    =    \(2^{3} \cdot 97\)
Sign: $0.544 + 0.838i$
Analytic conductor: \(0.387274\)
Root analytic conductor: \(0.622313\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{776} (243, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 776,\ (\ :0),\ 0.544 + 0.838i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5599064354\)
\(L(\frac12)\) \(\approx\) \(0.5599064354\)
\(L(1)\) 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 \( 1 + (0.128 + 1.95i)T + (-0.991 + 0.130i)T^{2} \)
19 \( 1 + (-0.923 - 1.38i)T + (-0.382 + 0.923i)T^{2} \)
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 \( 1 + (-1.05 + 0.357i)T + (0.793 - 0.608i)T^{2} \)
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 \( 1 + (0.257 + 0.293i)T + (-0.130 + 0.991i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (1.57 - 1.05i)T + (0.382 - 0.923i)T^{2} \)
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 \( 1 + (-1.50 + 0.0983i)T + (0.991 - 0.130i)T^{2} \)
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

−10.19980245649025209450948004992, −9.442463585223579853980033185878, −8.836542158787169991106851209426, −7.77101992329132053386397731921, −7.09282281981598630677924475530, −6.10467283699295619237594059564, −5.40771152443447783738500654489, −3.68711405428736212970494111200, −2.54786791482735560708900481277, −0.870998555614100719394191661839, 1.65146817030295823442688008730, 2.83609896731445773924294241840, 4.22178270592866691028816896460, 5.48932478279662890670072381319, 6.41077777512137045224751192975, 7.48543291373176012880861906032, 8.056426685838551973604881383726, 9.071392100539917671025260628006, 9.746872484618565673066908786336, 10.75705073910466996311720329355

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