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Properties

Label 2-776-776.363-c0-0-0
Degree $2$
Conductor $776$
Sign $0.188 + 0.982i$
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  + (−0.608 + 0.793i)2-s + (−0.207 − 1.57i)3-s + (−0.258 − 0.965i)4-s + (1.37 + 0.793i)6-s + (0.923 + 0.382i)8-s + (−1.46 + 0.392i)9-s + (1.53 − 1.17i)11-s + (−1.46 + 0.607i)12-s + (−0.866 + 0.499i)16-s + (0.172 + 0.349i)17-s + (0.580 − 1.40i)18-s + (−0.923 − 0.617i)19-s + 1.93i·22-s + (0.410 − 1.53i)24-s + (−0.991 − 0.130i)25-s + ⋯
L(s)  = 1  + (−0.608 + 0.793i)2-s + (−0.207 − 1.57i)3-s + (−0.258 − 0.965i)4-s + (1.37 + 0.793i)6-s + (0.923 + 0.382i)8-s + (−1.46 + 0.392i)9-s + (1.53 − 1.17i)11-s + (−1.46 + 0.607i)12-s + (−0.866 + 0.499i)16-s + (0.172 + 0.349i)17-s + (0.580 − 1.40i)18-s + (−0.923 − 0.617i)19-s + 1.93i·22-s + (0.410 − 1.53i)24-s + (−0.991 − 0.130i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 776 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.188 + 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 776 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.188 + 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(776\)    =    \(2^{3} \cdot 97\)
Sign: $0.188 + 0.982i$
Analytic conductor: \(0.387274\)
Root analytic conductor: \(0.622313\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{776} (363, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 776,\ (\ :0),\ 0.188 + 0.982i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6489804095\)
\(L(\frac12)\) \(\approx\) \(0.6489804095\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.608 - 0.793i)T \)
97 \( 1 + (-0.130 - 0.991i)T \)
good3 \( 1 + (0.207 + 1.57i)T + (-0.965 + 0.258i)T^{2} \)
5 \( 1 + (0.991 + 0.130i)T^{2} \)
7 \( 1 + (-0.608 - 0.793i)T^{2} \)
11 \( 1 + (-1.53 + 1.17i)T + (0.258 - 0.965i)T^{2} \)
13 \( 1 + (-0.991 - 0.130i)T^{2} \)
17 \( 1 + (-0.172 - 0.349i)T + (-0.608 + 0.793i)T^{2} \)
19 \( 1 + (0.923 + 0.617i)T + (0.382 + 0.923i)T^{2} \)
23 \( 1 + (-0.793 - 0.608i)T^{2} \)
29 \( 1 + (-0.130 + 0.991i)T^{2} \)
31 \( 1 + (-0.965 + 0.258i)T^{2} \)
37 \( 1 + (0.793 - 0.608i)T^{2} \)
41 \( 1 + (1.09 + 1.25i)T + (-0.130 + 0.991i)T^{2} \)
43 \( 1 + (-0.252 - 0.0675i)T + (0.866 + 0.5i)T^{2} \)
47 \( 1 + iT^{2} \)
53 \( 1 + (-0.258 - 0.965i)T^{2} \)
59 \( 1 + (-1.85 + 0.630i)T + (0.793 - 0.608i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.732 - 1.09i)T + (-0.382 - 0.923i)T^{2} \)
71 \( 1 + (0.130 + 0.991i)T^{2} \)
73 \( 1 + (0.366 - 1.36i)T + (-0.866 - 0.5i)T^{2} \)
79 \( 1 + (0.707 + 0.707i)T^{2} \)
83 \( 1 + (-0.576 + 0.284i)T + (0.608 - 0.793i)T^{2} \)
89 \( 1 + (-1.83 - 0.758i)T + (0.707 + 0.707i)T^{2} \)
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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

−10.27292795166795158747539620747, −9.003970989726991877179299553112, −8.545053135196494413244182483094, −7.66253732855794243317482937069, −6.77467850177588442849788773841, −6.29830228464089078851044063160, −5.54601044735654670584382234906, −3.94929537441536422994531176965, −2.08121939136187110948610532022, −0.919627321279888668156759251419, 1.89836678669043984228881747410, 3.48456566326930397956731790828, 4.14295950804549159209536930958, 4.87402554046270084383632119168, 6.30897768639054761381593512023, 7.45371740587227324721774417961, 8.655397106019207504865041362722, 9.270715496170800130851985710862, 10.00285605982799474079147891628, 10.34293767490929962986066048838

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