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Properties

Label 2-936-117.61-c1-0-28
Degree $2$
Conductor $936$
Sign $-0.911 + 0.410i$
Analytic cond. $7.47399$
Root an. cond. $2.73386$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.813 − 1.52i)3-s + (0.950 + 1.64i)5-s − 2.76·7-s + (−1.67 + 2.48i)9-s + (0.759 + 1.31i)11-s + (3.30 − 1.44i)13-s + (1.74 − 2.79i)15-s + (−1.83 − 3.17i)17-s + (−3.58 − 6.21i)19-s + (2.24 + 4.22i)21-s − 6.96·23-s + (0.694 − 1.20i)25-s + (5.16 + 0.543i)27-s + (−2.66 − 4.60i)29-s + (−4.05 − 7.02i)31-s + ⋯
L(s)  = 1  + (−0.469 − 0.882i)3-s + (0.424 + 0.735i)5-s − 1.04·7-s + (−0.559 + 0.828i)9-s + (0.228 + 0.396i)11-s + (0.916 − 0.400i)13-s + (0.450 − 0.720i)15-s + (−0.444 − 0.769i)17-s + (−0.823 − 1.42i)19-s + (0.490 + 0.922i)21-s − 1.45·23-s + (0.138 − 0.240i)25-s + (0.994 + 0.104i)27-s + (−0.493 − 0.855i)29-s + (−0.728 − 1.26i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.911 + 0.410i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.911 + 0.410i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(936\)    =    \(2^{3} \cdot 3^{2} \cdot 13\)
Sign: $-0.911 + 0.410i$
Analytic conductor: \(7.47399\)
Root analytic conductor: \(2.73386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{936} (529, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 936,\ (\ :1/2),\ -0.911 + 0.410i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.108724 - 0.506188i\)
\(L(\frac12)\) \(\approx\) \(0.108724 - 0.506188i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (0.813 + 1.52i)T \)
13 \( 1 + (-3.30 + 1.44i)T \)
good5 \( 1 + (-0.950 - 1.64i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + 2.76T + 7T^{2} \)
11 \( 1 + (-0.759 - 1.31i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (1.83 + 3.17i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.58 + 6.21i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 6.96T + 23T^{2} \)
29 \( 1 + (2.66 + 4.60i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (4.05 + 7.02i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (4.41 - 7.65i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 1.81T + 41T^{2} \)
43 \( 1 + 0.791T + 43T^{2} \)
47 \( 1 + (0.382 - 0.661i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 6.15T + 53T^{2} \)
59 \( 1 + (0.779 - 1.34i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + 13.1T + 61T^{2} \)
67 \( 1 + 1.10T + 67T^{2} \)
71 \( 1 + (-7.35 - 12.7i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 1.28T + 73T^{2} \)
79 \( 1 + (-2.49 + 4.32i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-4.82 + 8.35i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-7.48 + 12.9i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 13.6T + 97T^{2} \)
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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

−9.795495300897277833757889895681, −8.885978675401472907713075529841, −7.83475127157397612033078281475, −6.86346337436781547610964874301, −6.40366580732957868296332070763, −5.72378039106887631849502024404, −4.32746371330521104890149609151, −2.95583070122369960621680008706, −2.08184803495466863221551483533, −0.24602625007944148615389993479, 1.66055166281792815461491133897, 3.56946260383317137437379905835, 3.97447118228547486093934340143, 5.33664250022843803689669744210, 6.03216884466656196034471842309, 6.62273526554293110097018456517, 8.237954704577970565512101546030, 9.003707279231329294250328378070, 9.469574313624572198389067285445, 10.57309935640666022502297601327

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