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Properties

Label 2-776-776.723-c0-0-0
Degree $2$
Conductor $776$
Sign $0.946 - 0.321i$
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  + (0.130 − 0.991i)2-s + (1.20 + 1.57i)3-s + (−0.965 − 0.258i)4-s + (1.71 − 0.991i)6-s + (−0.382 + 0.923i)8-s + (−0.758 + 2.83i)9-s + (−0.513 + 0.0675i)11-s + (−0.758 − 1.83i)12-s + (0.866 + 0.5i)16-s + (1.25 − 1.09i)17-s + (2.70 + 1.12i)18-s + (0.382 − 0.0761i)19-s + 0.517i·22-s + (−1.91 + 0.513i)24-s + (−0.793 − 0.608i)25-s + ⋯
L(s)  = 1  + (0.130 − 0.991i)2-s + (1.20 + 1.57i)3-s + (−0.965 − 0.258i)4-s + (1.71 − 0.991i)6-s + (−0.382 + 0.923i)8-s + (−0.758 + 2.83i)9-s + (−0.513 + 0.0675i)11-s + (−0.758 − 1.83i)12-s + (0.866 + 0.5i)16-s + (1.25 − 1.09i)17-s + (2.70 + 1.12i)18-s + (0.382 − 0.0761i)19-s + 0.517i·22-s + (−1.91 + 0.513i)24-s + (−0.793 − 0.608i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 776 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.946 - 0.321i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 776 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.946 - 0.321i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(776\)    =    \(2^{3} \cdot 97\)
Sign: $0.946 - 0.321i$
Analytic conductor: \(0.387274\)
Root analytic conductor: \(0.622313\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{776} (723, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 776,\ (\ :0),\ 0.946 - 0.321i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.303673191\)
\(L(\frac12)\) \(\approx\) \(1.303673191\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.130 + 0.991i)T \)
97 \( 1 + (-0.608 - 0.793i)T \)
good3 \( 1 + (-1.20 - 1.57i)T + (-0.258 + 0.965i)T^{2} \)
5 \( 1 + (0.793 + 0.608i)T^{2} \)
7 \( 1 + (0.130 + 0.991i)T^{2} \)
11 \( 1 + (0.513 - 0.0675i)T + (0.965 - 0.258i)T^{2} \)
13 \( 1 + (-0.793 - 0.608i)T^{2} \)
17 \( 1 + (-1.25 + 1.09i)T + (0.130 - 0.991i)T^{2} \)
19 \( 1 + (-0.382 + 0.0761i)T + (0.923 - 0.382i)T^{2} \)
23 \( 1 + (0.991 + 0.130i)T^{2} \)
29 \( 1 + (-0.608 + 0.793i)T^{2} \)
31 \( 1 + (-0.258 + 0.965i)T^{2} \)
37 \( 1 + (-0.991 + 0.130i)T^{2} \)
41 \( 1 + (0.867 + 1.75i)T + (-0.608 + 0.793i)T^{2} \)
43 \( 1 + (-0.315 - 1.17i)T + (-0.866 + 0.5i)T^{2} \)
47 \( 1 + iT^{2} \)
53 \( 1 + (-0.965 - 0.258i)T^{2} \)
59 \( 1 + (0.0726 + 1.10i)T + (-0.991 + 0.130i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (-0.172 - 0.867i)T + (-0.923 + 0.382i)T^{2} \)
71 \( 1 + (0.608 + 0.793i)T^{2} \)
73 \( 1 + (-1.36 + 0.366i)T + (0.866 - 0.5i)T^{2} \)
79 \( 1 + (-0.707 - 0.707i)T^{2} \)
83 \( 1 + (1.31 + 1.50i)T + (-0.130 + 0.991i)T^{2} \)
89 \( 1 + (0.607 - 1.46i)T + (-0.707 - 0.707i)T^{2} \)
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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

−10.26419545755612285502702465174, −9.855876463430703472625716643268, −9.199735775498950746054380934462, −8.326122515414339660142463895502, −7.63445210424226560437519878999, −5.47139190334160887457972022307, −4.91112194689513734036939168284, −3.86455166031266419434542295470, −3.14046627796429549838669434169, −2.24363234608343360947470994504, 1.42097680109733257138346677014, 2.98091723723471126131280703385, 3.83578745217004689351288114665, 5.55077990196257837967394644596, 6.28673984652543889179977464705, 7.22573782195552208970458399857, 7.87520499923597795849030347213, 8.313415857660033166330768016606, 9.241158709959409886590472968101, 10.04433545372026216536605983008

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