Properties

Label 2-772-772.255-c0-0-0
Degree $2$
Conductor $772$
Sign $-0.763 + 0.646i$
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  + (0.130 − 0.991i)2-s + (−0.965 − 0.258i)4-s + (0.534 − 1.57i)5-s + (−0.382 + 0.923i)8-s i·9-s + (−1.49 − 0.735i)10-s + (0.735 + 0.491i)13-s + (0.866 + 0.5i)16-s + (−1.29 + 1.47i)17-s + (−0.991 − 0.130i)18-s + (−0.923 + 1.38i)20-s + (−1.40 − 1.07i)25-s + (0.583 − 0.665i)26-s + (−0.996 − 1.49i)29-s + (0.608 − 0.793i)32-s + ⋯
L(s)  = 1  + (0.130 − 0.991i)2-s + (−0.965 − 0.258i)4-s + (0.534 − 1.57i)5-s + (−0.382 + 0.923i)8-s i·9-s + (−1.49 − 0.735i)10-s + (0.735 + 0.491i)13-s + (0.866 + 0.5i)16-s + (−1.29 + 1.47i)17-s + (−0.991 − 0.130i)18-s + (−0.923 + 1.38i)20-s + (−1.40 − 1.07i)25-s + (0.583 − 0.665i)26-s + (−0.996 − 1.49i)29-s + (0.608 − 0.793i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 772 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.763 + 0.646i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 772 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.763 + 0.646i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(772\)    =    \(2^{2} \cdot 193\)
Sign: $-0.763 + 0.646i$
Analytic conductor: \(0.385278\)
Root analytic conductor: \(0.620707\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{772} (255, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 772,\ (\ :0),\ -0.763 + 0.646i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9791625606\)
\(L(\frac12)\) \(\approx\) \(0.9791625606\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.130 + 0.991i)T \)
193 \( 1 + (0.707 - 0.707i)T \)
good3 \( 1 + iT^{2} \)
5 \( 1 + (-0.534 + 1.57i)T + (-0.793 - 0.608i)T^{2} \)
7 \( 1 + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.923 + 0.382i)T^{2} \)
13 \( 1 + (-0.735 - 0.491i)T + (0.382 + 0.923i)T^{2} \)
17 \( 1 + (1.29 - 1.47i)T + (-0.130 - 0.991i)T^{2} \)
19 \( 1 + (0.793 + 0.608i)T^{2} \)
23 \( 1 + (-0.707 + 0.707i)T^{2} \)
29 \( 1 + (0.996 + 1.49i)T + (-0.382 + 0.923i)T^{2} \)
31 \( 1 + (0.965 + 0.258i)T^{2} \)
37 \( 1 + (-1.88 - 0.123i)T + (0.991 + 0.130i)T^{2} \)
41 \( 1 + (-0.0983 - 0.0862i)T + (0.130 + 0.991i)T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + (0.991 + 0.130i)T^{2} \)
53 \( 1 + (0.0983 - 1.50i)T + (-0.991 - 0.130i)T^{2} \)
59 \( 1 + (0.866 - 0.5i)T^{2} \)
61 \( 1 + (-0.641 - 1.88i)T + (-0.793 + 0.608i)T^{2} \)
67 \( 1 + (-0.707 + 0.707i)T^{2} \)
71 \( 1 + (-0.923 + 0.382i)T^{2} \)
73 \( 1 + (-1.31 + 0.0862i)T + (0.991 - 0.130i)T^{2} \)
79 \( 1 + (-0.793 + 0.608i)T^{2} \)
83 \( 1 + (0.258 + 0.965i)T^{2} \)
89 \( 1 + (0.923 - 0.617i)T + (0.382 - 0.923i)T^{2} \)
97 \( 1 + (0.0675 + 0.513i)T + (-0.965 + 0.258i)T^{2} \)
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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

−10.09511853210015751137610734848, −9.257146429404697435112226513179, −8.876548989838852650036656968018, −8.096387425556686563034044583102, −6.25806240902220002330193067062, −5.69875976825484371740069226184, −4.33645365482320355539558829347, −3.99131723526020536489434513401, −2.19888845135035083034594986170, −1.10002274406697028666163995662, 2.40991930193802078733944442266, 3.45664185039781494903307179776, 4.80320623565642629346082343269, 5.71679256196430246747008714727, 6.62057390991197871894932295681, 7.21905591817230028408200172604, 8.019891652489596304233308733427, 9.129822099652981403638813921149, 9.891224355873371098678576825492, 10.93357480717117899671892780437

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