Properties

Label 2-772-772.627-c0-0-0
Degree $2$
Conductor $772$
Sign $0.978 + 0.204i$
Analytic cond. $0.385278$
Root an. cond. $0.620707$
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 + (0.965 − 0.258i)4-s + (−0.349 − 0.172i)5-s + (0.923 − 0.382i)8-s + i·9-s + (−0.369 − 0.125i)10-s + (−0.125 + 0.630i)13-s + (0.866 − 0.5i)16-s + (0.108 − 1.65i)17-s + (0.130 + 0.991i)18-s + (−0.382 − 0.0761i)20-s + (−0.516 − 0.672i)25-s + (−0.0420 + 0.641i)26-s + (−1.85 + 0.369i)29-s + (0.793 − 0.608i)32-s + ⋯
L(s)  = 1  + (0.991 − 0.130i)2-s + (0.965 − 0.258i)4-s + (−0.349 − 0.172i)5-s + (0.923 − 0.382i)8-s + i·9-s + (−0.369 − 0.125i)10-s + (−0.125 + 0.630i)13-s + (0.866 − 0.5i)16-s + (0.108 − 1.65i)17-s + (0.130 + 0.991i)18-s + (−0.382 − 0.0761i)20-s + (−0.516 − 0.672i)25-s + (−0.0420 + 0.641i)26-s + (−1.85 + 0.369i)29-s + (0.793 − 0.608i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 772 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.978 + 0.204i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 772 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.978 + 0.204i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(772\)    =    \(2^{2} \cdot 193\)
Sign: $0.978 + 0.204i$
Analytic conductor: \(0.385278\)
Root analytic conductor: \(0.620707\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{772} (627, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 772,\ (\ :0),\ 0.978 + 0.204i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.647167030\)
\(L(\frac12)\) \(\approx\) \(1.647167030\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.991 + 0.130i)T \)
193 \( 1 + (-0.707 - 0.707i)T \)
good3 \( 1 - iT^{2} \)
5 \( 1 + (0.349 + 0.172i)T + (0.608 + 0.793i)T^{2} \)
7 \( 1 + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.382 + 0.923i)T^{2} \)
13 \( 1 + (0.125 - 0.630i)T + (-0.923 - 0.382i)T^{2} \)
17 \( 1 + (-0.108 + 1.65i)T + (-0.991 - 0.130i)T^{2} \)
19 \( 1 + (-0.608 - 0.793i)T^{2} \)
23 \( 1 + (0.707 + 0.707i)T^{2} \)
29 \( 1 + (1.85 - 0.369i)T + (0.923 - 0.382i)T^{2} \)
31 \( 1 + (-0.965 + 0.258i)T^{2} \)
37 \( 1 + (0.583 - 0.665i)T + (-0.130 - 0.991i)T^{2} \)
41 \( 1 + (1.50 + 0.0983i)T + (0.991 + 0.130i)T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + (-0.130 - 0.991i)T^{2} \)
53 \( 1 + (-1.50 - 1.31i)T + (0.130 + 0.991i)T^{2} \)
59 \( 1 + (0.866 + 0.5i)T^{2} \)
61 \( 1 + (1.18 - 0.583i)T + (0.608 - 0.793i)T^{2} \)
67 \( 1 + (0.707 + 0.707i)T^{2} \)
71 \( 1 + (-0.382 + 0.923i)T^{2} \)
73 \( 1 + (-0.0862 - 0.0983i)T + (-0.130 + 0.991i)T^{2} \)
79 \( 1 + (0.608 - 0.793i)T^{2} \)
83 \( 1 + (-0.258 + 0.965i)T^{2} \)
89 \( 1 + (0.382 + 1.92i)T + (-0.923 + 0.382i)T^{2} \)
97 \( 1 + (-0.513 - 0.0675i)T + (0.965 + 0.258i)T^{2} \)
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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

−10.70085541386444839569093388497, −9.851325931656069275692730653310, −8.769932620879296922752366365681, −7.53289972853495834971682040044, −7.13641703065656690712189782773, −5.83672368587621120841898220641, −4.98712762436456591266089079920, −4.24748493943247380572428918156, −3.01860628534416485045486503224, −1.89005802164656225508777371854, 1.90924306303309231507689577665, 3.53405666006800496943079287722, 3.81868230976407912583436986179, 5.30221908640517079152443423893, 6.02254488355715884306631878604, 6.94043664043590038848562287548, 7.77297984791195316789458147068, 8.679963264952983189900844088611, 9.899826814389973386788085688870, 10.73331643169922461569523221900

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