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Properties

Label 2-772-772.239-c0-0-0
Degree $2$
Conductor $772$
Sign $-0.989 + 0.145i$
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  + (0.608 − 0.793i)2-s + (−0.258 − 0.965i)4-s + (−1.95 − 0.128i)5-s + (−0.923 − 0.382i)8-s − i·9-s + (−1.29 + 1.47i)10-s + (−1.47 + 0.293i)13-s + (−0.866 + 0.499i)16-s + (0.491 − 0.996i)17-s + (−0.793 − 0.608i)18-s + (0.382 + 1.92i)20-s + (2.82 + 0.371i)25-s + (−0.665 + 1.34i)26-s + (0.257 − 1.29i)29-s + (−0.130 + 0.991i)32-s + ⋯
L(s)  = 1  + (0.608 − 0.793i)2-s + (−0.258 − 0.965i)4-s + (−1.95 − 0.128i)5-s + (−0.923 − 0.382i)8-s − i·9-s + (−1.29 + 1.47i)10-s + (−1.47 + 0.293i)13-s + (−0.866 + 0.499i)16-s + (0.491 − 0.996i)17-s + (−0.793 − 0.608i)18-s + (0.382 + 1.92i)20-s + (2.82 + 0.371i)25-s + (−0.665 + 1.34i)26-s + (0.257 − 1.29i)29-s + (−0.130 + 0.991i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 772 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.989 + 0.145i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 772 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.989 + 0.145i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(772\)    =    \(2^{2} \cdot 193\)
Sign: $-0.989 + 0.145i$
Analytic conductor: \(0.385278\)
Root analytic conductor: \(0.620707\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{772} (239, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 772,\ (\ :0),\ -0.989 + 0.145i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6485041679\)
\(L(\frac12)\) \(\approx\) \(0.6485041679\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.608 + 0.793i)T \)
193 \( 1 + (-0.707 + 0.707i)T \)
good3 \( 1 + iT^{2} \)
5 \( 1 + (1.95 + 0.128i)T + (0.991 + 0.130i)T^{2} \)
7 \( 1 + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (-0.382 + 0.923i)T^{2} \)
13 \( 1 + (1.47 - 0.293i)T + (0.923 - 0.382i)T^{2} \)
17 \( 1 + (-0.491 + 0.996i)T + (-0.608 - 0.793i)T^{2} \)
19 \( 1 + (-0.991 - 0.130i)T^{2} \)
23 \( 1 + (0.707 - 0.707i)T^{2} \)
29 \( 1 + (-0.257 + 1.29i)T + (-0.923 - 0.382i)T^{2} \)
31 \( 1 + (0.258 + 0.965i)T^{2} \)
37 \( 1 + (0.123 + 0.0420i)T + (0.793 + 0.608i)T^{2} \)
41 \( 1 + (0.576 + 0.284i)T + (0.608 + 0.793i)T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + (0.793 + 0.608i)T^{2} \)
53 \( 1 + (-0.576 + 1.69i)T + (-0.793 - 0.608i)T^{2} \)
59 \( 1 + (-0.866 - 0.5i)T^{2} \)
61 \( 1 + (-1.88 + 0.123i)T + (0.991 - 0.130i)T^{2} \)
67 \( 1 + (0.707 - 0.707i)T^{2} \)
71 \( 1 + (0.382 + 0.923i)T^{2} \)
73 \( 1 + (0.837 - 0.284i)T + (0.793 - 0.608i)T^{2} \)
79 \( 1 + (0.991 - 0.130i)T^{2} \)
83 \( 1 + (0.965 + 0.258i)T^{2} \)
89 \( 1 + (-0.382 - 0.0761i)T + (0.923 + 0.382i)T^{2} \)
97 \( 1 + (1.17 + 1.53i)T + (-0.258 + 0.965i)T^{2} \)
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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

−10.18833128616223253171461024776, −9.448723639760206253412370473315, −8.494286587825622054445197293855, −7.42481492387616038751121159811, −6.73629902379430423999341626852, −5.25342213249157445279213567878, −4.40141231384825957633808200528, −3.66605820088879366145892144257, −2.70454663379331532259835206004, −0.55502463297254491571726824398, 2.79033598182948095848059520565, 3.79376670651778575420820456753, 4.65812154867556915244493717171, 5.38368926889529977783716915113, 6.90579300113384736948760825656, 7.44978551137690282487673409253, 8.091116381995662327758754591858, 8.715923012607332276720363891398, 10.24052962053118431015979768973, 11.07745864029709336112241689287

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