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Properties

Label 2-936-936.155-c1-0-110
Degree $2$
Conductor $936$
Sign $0.0454 + 0.998i$
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  + (1.22 + 0.707i)2-s + (−1.02 − 1.39i)3-s + (0.999 + 1.73i)4-s + (−3.74 + 2.16i)5-s + (−0.268 − 2.43i)6-s + (1.88 − 3.27i)7-s + 2.82i·8-s + (−0.896 + 2.86i)9-s − 6.11·10-s + (1.39 − 3.17i)12-s + (−1.80 − 3.12i)13-s + (4.62 − 2.67i)14-s + (6.85 + 3.00i)15-s + (−2.00 + 3.46i)16-s − 7.65i·17-s + (−3.12 + 2.87i)18-s + ⋯
L(s)  = 1  + (0.866 + 0.499i)2-s + (−0.592 − 0.805i)3-s + (0.499 + 0.866i)4-s + (−1.67 + 0.966i)5-s + (−0.109 − 0.993i)6-s + (0.714 − 1.23i)7-s + 0.999i·8-s + (−0.298 + 0.954i)9-s − 1.93·10-s + (0.401 − 0.915i)12-s + (−0.499 − 0.866i)13-s + (1.23 − 0.714i)14-s + (1.76 + 0.776i)15-s + (−0.500 + 0.866i)16-s − 1.85i·17-s + (−0.736 + 0.676i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0454 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0454 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(936\)    =    \(2^{3} \cdot 3^{2} \cdot 13\)
Sign: $0.0454 + 0.998i$
Analytic conductor: \(7.47399\)
Root analytic conductor: \(2.73386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{936} (155, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 936,\ (\ :1/2),\ 0.0454 + 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.731117 - 0.698637i\)
\(L(\frac12)\) \(\approx\) \(0.731117 - 0.698637i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.22 - 0.707i)T \)
3 \( 1 + (1.02 + 1.39i)T \)
13 \( 1 + (1.80 + 3.12i)T \)
good5 \( 1 + (3.74 - 2.16i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 + (-1.88 + 3.27i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + 7.65iT - 17T^{2} \)
19 \( 1 - 19T^{2} \)
23 \( 1 + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (1.87 + 3.24i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 6.74T + 37T^{2} \)
41 \( 1 + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-6.55 + 11.3i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (7.84 + 4.53i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 15.7iT - 71T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 + (48.5 + 84.0i)T^{2} \)
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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

−10.38790089218902124991722239008, −8.366787716155839038128824368923, −7.66204695460257907740293569177, −7.20388184697435830171274337374, −6.86761558793987865992878743921, −5.38521414977444568220075607195, −4.58983804596465341777164199904, −3.64011306115713376810519258136, −2.60547563349036427776102408327, −0.38898890031961675364351200062, 1.57462401818568846977781755262, 3.31186603858812431209452892052, 4.23291762232390610106789670808, 4.75591535388826817391803594901, 5.51334349938975112565076706797, 6.52421102733735036304859714000, 7.88163879663951777252882189560, 8.743565190009146567894852305890, 9.398186108372479839335954685068, 10.70009123758353995688169194913

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