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Properties

Label 2-156-52.15-c1-0-10
Degree $2$
Conductor $156$
Sign $0.996 + 0.0878i$
Analytic cond. $1.24566$
Root an. cond. $1.11609$
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  + (1.41 − 0.0965i)2-s + (0.866 − 0.5i)3-s + (1.98 − 0.272i)4-s + (−1.91 + 1.91i)5-s + (1.17 − 0.789i)6-s + (0.104 + 0.390i)7-s + (2.76 − 0.575i)8-s + (0.499 − 0.866i)9-s + (−2.52 + 2.89i)10-s + (−4.57 − 1.22i)11-s + (1.57 − 1.22i)12-s + (0.711 − 3.53i)13-s + (0.185 + 0.540i)14-s + (−0.702 + 2.62i)15-s + (3.85 − 1.07i)16-s + (−1.46 − 0.848i)17-s + ⋯
L(s)  = 1  + (0.997 − 0.0682i)2-s + (0.499 − 0.288i)3-s + (0.990 − 0.136i)4-s + (−0.858 + 0.858i)5-s + (0.479 − 0.322i)6-s + (0.0395 + 0.147i)7-s + (0.979 − 0.203i)8-s + (0.166 − 0.288i)9-s + (−0.797 + 0.914i)10-s + (−1.37 − 0.369i)11-s + (0.456 − 0.354i)12-s + (0.197 − 0.980i)13-s + (0.0494 + 0.144i)14-s + (−0.181 + 0.676i)15-s + (0.962 − 0.269i)16-s + (−0.356 − 0.205i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 + 0.0878i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.996 + 0.0878i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(156\)    =    \(2^{2} \cdot 3 \cdot 13\)
Sign: $0.996 + 0.0878i$
Analytic conductor: \(1.24566\)
Root analytic conductor: \(1.11609\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{156} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 156,\ (\ :1/2),\ 0.996 + 0.0878i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.95073 - 0.0858123i\)
\(L(\frac12)\) \(\approx\) \(1.95073 - 0.0858123i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.41 + 0.0965i)T \)
3 \( 1 + (-0.866 + 0.5i)T \)
13 \( 1 + (-0.711 + 3.53i)T \)
good5 \( 1 + (1.91 - 1.91i)T - 5iT^{2} \)
7 \( 1 + (-0.104 - 0.390i)T + (-6.06 + 3.5i)T^{2} \)
11 \( 1 + (4.57 + 1.22i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (1.46 + 0.848i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (5.62 - 1.50i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (-1.13 - 1.97i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.57 - 6.19i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-6.14 - 6.14i)T + 31iT^{2} \)
37 \( 1 + (-1.60 + 5.97i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (4.82 + 1.29i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (-2.78 + 4.82i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (4.50 - 4.50i)T - 47iT^{2} \)
53 \( 1 + 0.959T + 53T^{2} \)
59 \( 1 + (2.59 + 9.68i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (2.02 - 3.51i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.04 - 3.89i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (-11.1 + 2.99i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (4.41 + 4.41i)T + 73iT^{2} \)
79 \( 1 + 2.09iT - 79T^{2} \)
83 \( 1 + (-9.94 - 9.94i)T + 83iT^{2} \)
89 \( 1 + (-2.48 + 9.27i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (-4.73 - 17.6i)T + (-84.0 + 48.5i)T^{2} \)
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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

−12.95842234320497826545462521744, −12.19044387523020197778865594714, −10.87597755758861086993977914684, −10.48691622139953740374845288372, −8.387292667713350527231852022293, −7.56327050422993604776258295653, −6.52405749381964380478351385468, −5.12703387945270101999947740345, −3.54811454745479835931530621950, −2.63524103859778847338147616934, 2.48395584539505468186370650099, 4.25481316291646386168700840732, 4.70619757668721843484263133549, 6.42769806734483233107381424081, 7.81316586579840594157302300834, 8.502113327439698878120601459131, 10.09897465523908534133152898726, 11.18953185324414114094580895388, 12.16127413966382166848022173423, 13.09181431535819327037195831518

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