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Properties

Label 2-936-117.61-c1-0-38
Degree $2$
Conductor $936$
Sign $-0.741 + 0.670i$
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  + (1.31 − 1.12i)3-s + (0.766 + 1.32i)5-s − 3.51·7-s + (0.456 − 2.96i)9-s + (−3.19 − 5.53i)11-s + (−3.46 + 1.00i)13-s + (2.50 + 0.881i)15-s + (−0.550 − 0.953i)17-s + (−1.13 − 1.97i)19-s + (−4.62 + 3.96i)21-s − 3.32·23-s + (1.32 − 2.29i)25-s + (−2.74 − 4.41i)27-s + (0.714 + 1.23i)29-s + (−1.93 − 3.35i)31-s + ⋯
L(s)  = 1  + (0.759 − 0.651i)3-s + (0.342 + 0.594i)5-s − 1.32·7-s + (0.152 − 0.988i)9-s + (−0.963 − 1.66i)11-s + (−0.960 + 0.279i)13-s + (0.647 + 0.227i)15-s + (−0.133 − 0.231i)17-s + (−0.261 − 0.452i)19-s + (−1.00 + 0.864i)21-s − 0.693·23-s + (0.264 − 0.458i)25-s + (−0.527 − 0.849i)27-s + (0.132 + 0.229i)29-s + (−0.347 − 0.602i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.741 + 0.670i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.741 + 0.670i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(936\)    =    \(2^{3} \cdot 3^{2} \cdot 13\)
Sign: $-0.741 + 0.670i$
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: \((2,\ 936,\ (\ :1/2),\ -0.741 + 0.670i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.383276 - 0.995395i\)
\(L(\frac12)\) \(\approx\) \(0.383276 - 0.995395i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.31 + 1.12i)T \)
13 \( 1 + (3.46 - 1.00i)T \)
good5 \( 1 + (-0.766 - 1.32i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + 3.51T + 7T^{2} \)
11 \( 1 + (3.19 + 5.53i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (0.550 + 0.953i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.13 + 1.97i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 3.32T + 23T^{2} \)
29 \( 1 + (-0.714 - 1.23i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (1.93 + 3.35i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.81 + 3.14i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 11.9T + 41T^{2} \)
43 \( 1 - 12.5T + 43T^{2} \)
47 \( 1 + (1.23 - 2.13i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 6.18T + 53T^{2} \)
59 \( 1 + (2.42 - 4.20i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 - 3.09T + 61T^{2} \)
67 \( 1 + 7.60T + 67T^{2} \)
71 \( 1 + (-2.43 - 4.21i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 5.40T + 73T^{2} \)
79 \( 1 + (-2.39 + 4.14i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (3.19 - 5.53i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (4.25 - 7.36i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 3.52T + 97T^{2} \)
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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

−9.564998085250022783554204557582, −9.023264135665716559474185840474, −7.959715641492360611815938728622, −7.23659449150107604004833407149, −6.30629064941983355984783428372, −5.79009895363937037233255718799, −4.06996460097633242291484424212, −2.81698133063629165187035690839, −2.60918077390696204638208995190, −0.41311794464140060597234835854, 2.08962276181446484388151544178, 2.94320620994195900377913249255, 4.20601102908912950012983445238, 4.94373398073363584322460247964, 5.95023762367628813805644482419, 7.24937285675037737664447393477, 7.81754653788464074456798776906, 8.986368380834824480893961125900, 9.689434430055933638202511665964, 9.992007377915868168781219042887

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