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Properties

Label 2-776-776.539-c0-0-0
Degree $2$
Conductor $776$
Sign $-0.988 - 0.150i$
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  + (0.258 + 0.965i)2-s + (−0.5 + 0.133i)3-s + (−0.866 + 0.499i)4-s + (−0.258 − 0.448i)6-s + (−0.707 − 0.707i)8-s + (−0.633 + 0.366i)9-s + (−0.448 + 1.67i)11-s + (0.366 − 0.366i)12-s + (0.500 − 0.866i)16-s + (0.607 + 0.465i)17-s + (−0.517 − 0.517i)18-s + (−1.70 + 0.707i)19-s − 1.73·22-s + (0.448 + 0.258i)24-s + (−0.965 − 0.258i)25-s + ⋯
L(s)  = 1  + (0.258 + 0.965i)2-s + (−0.5 + 0.133i)3-s + (−0.866 + 0.499i)4-s + (−0.258 − 0.448i)6-s + (−0.707 − 0.707i)8-s + (−0.633 + 0.366i)9-s + (−0.448 + 1.67i)11-s + (0.366 − 0.366i)12-s + (0.500 − 0.866i)16-s + (0.607 + 0.465i)17-s + (−0.517 − 0.517i)18-s + (−1.70 + 0.707i)19-s − 1.73·22-s + (0.448 + 0.258i)24-s + (−0.965 − 0.258i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 776 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.988 - 0.150i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 776 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.988 - 0.150i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(776\)    =    \(2^{3} \cdot 97\)
Sign: $-0.988 - 0.150i$
Analytic conductor: \(0.387274\)
Root analytic conductor: \(0.622313\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{776} (539, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 776,\ (\ :0),\ -0.988 - 0.150i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6159303363\)
\(L(\frac12)\) \(\approx\) \(0.6159303363\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.258 - 0.965i)T \)
97 \( 1 + (-0.965 + 0.258i)T \)
good3 \( 1 + (0.5 - 0.133i)T + (0.866 - 0.5i)T^{2} \)
5 \( 1 + (0.965 + 0.258i)T^{2} \)
7 \( 1 + (-0.258 + 0.965i)T^{2} \)
11 \( 1 + (0.448 - 1.67i)T + (-0.866 - 0.5i)T^{2} \)
13 \( 1 + (0.965 + 0.258i)T^{2} \)
17 \( 1 + (-0.607 - 0.465i)T + (0.258 + 0.965i)T^{2} \)
19 \( 1 + (1.70 - 0.707i)T + (0.707 - 0.707i)T^{2} \)
23 \( 1 + (0.258 + 0.965i)T^{2} \)
29 \( 1 + (-0.965 - 0.258i)T^{2} \)
31 \( 1 + (-0.866 + 0.5i)T^{2} \)
37 \( 1 + (0.258 - 0.965i)T^{2} \)
41 \( 1 + (-1.83 - 0.241i)T + (0.965 + 0.258i)T^{2} \)
43 \( 1 + (-1.67 - 0.965i)T + (0.5 + 0.866i)T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + (0.866 - 0.5i)T^{2} \)
59 \( 1 + (0.465 + 0.607i)T + (-0.258 + 0.965i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (1.83 - 0.758i)T + (0.707 - 0.707i)T^{2} \)
71 \( 1 + (-0.965 + 0.258i)T^{2} \)
73 \( 1 + (0.5 + 0.866i)T^{2} \)
79 \( 1 - iT^{2} \)
83 \( 1 + (-0.965 - 0.741i)T + (0.258 + 0.965i)T^{2} \)
89 \( 1 + (-1.36 - 1.36i)T + iT^{2} \)
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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

−10.73818409727594933778739849942, −10.05088531229595852296275932457, −9.126815958579719719589482102132, −8.037831727905725490350999759741, −7.57285033966932380270249165507, −6.35275262755267581849566263579, −5.76689320552756515461892229368, −4.72744558259182194716843405847, −4.03481393650677739636832972219, −2.34931219759139854850813734566, 0.61880480843703651093382163743, 2.48455882516805052692740620302, 3.42335981777669448746719898302, 4.55781201495843219272400564559, 5.87058753213018978890023400031, 5.97913435975849543813228924201, 7.67220622493348100173052151093, 8.765967120107860215228710268017, 9.189515313106483178959743360759, 10.60615728948549847803231219739

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