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Properties

Label 2-156-52.51-c2-0-12
Degree $2$
Conductor $156$
Sign $0.798 + 0.601i$
Analytic cond. $4.25069$
Root an. cond. $2.06172$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.0830 − 1.99i)2-s + 1.73i·3-s + (−3.98 + 0.332i)4-s + 0.451i·5-s + (3.46 − 0.143i)6-s + 9.24·7-s + (0.994 + 7.93i)8-s − 2.99·9-s + (0.901 − 0.0374i)10-s + 11.2·11-s + (−0.575 − 6.90i)12-s + (8.66 − 9.69i)13-s + (−0.768 − 18.4i)14-s − 0.781·15-s + (15.7 − 2.64i)16-s − 5.48·17-s + ⋯
L(s)  = 1  + (−0.0415 − 0.999i)2-s + 0.577i·3-s + (−0.996 + 0.0830i)4-s + 0.0902i·5-s + (0.576 − 0.0239i)6-s + 1.32·7-s + (0.124 + 0.992i)8-s − 0.333·9-s + (0.0901 − 0.00374i)10-s + 1.02·11-s + (−0.0479 − 0.575i)12-s + (0.666 − 0.745i)13-s + (−0.0548 − 1.31i)14-s − 0.0520·15-s + (0.986 − 0.165i)16-s − 0.322·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.798 + 0.601i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.798 + 0.601i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(156\)    =    \(2^{2} \cdot 3 \cdot 13\)
Sign: $0.798 + 0.601i$
Analytic conductor: \(4.25069\)
Root analytic conductor: \(2.06172\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{156} (103, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 156,\ (\ :1),\ 0.798 + 0.601i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.43972 - 0.481861i\)
\(L(\frac12)\) \(\approx\) \(1.43972 - 0.481861i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.0830 + 1.99i)T \)
3 \( 1 - 1.73iT \)
13 \( 1 + (-8.66 + 9.69i)T \)
good5 \( 1 - 0.451iT - 25T^{2} \)
7 \( 1 - 9.24T + 49T^{2} \)
11 \( 1 - 11.2T + 121T^{2} \)
17 \( 1 + 5.48T + 289T^{2} \)
19 \( 1 - 15.6T + 361T^{2} \)
23 \( 1 - 15.8iT - 529T^{2} \)
29 \( 1 + 41.4T + 841T^{2} \)
31 \( 1 + 13.7T + 961T^{2} \)
37 \( 1 + 1.35iT - 1.36e3T^{2} \)
41 \( 1 + 31.3iT - 1.68e3T^{2} \)
43 \( 1 - 78.9iT - 1.84e3T^{2} \)
47 \( 1 - 39.5T + 2.20e3T^{2} \)
53 \( 1 + 67.3T + 2.80e3T^{2} \)
59 \( 1 + 59.4T + 3.48e3T^{2} \)
61 \( 1 - 36.6T + 3.72e3T^{2} \)
67 \( 1 + 66.2T + 4.48e3T^{2} \)
71 \( 1 - 66.8T + 5.04e3T^{2} \)
73 \( 1 + 82.4iT - 5.32e3T^{2} \)
79 \( 1 + 91.4iT - 6.24e3T^{2} \)
83 \( 1 + 121.T + 6.88e3T^{2} \)
89 \( 1 - 69.6iT - 7.92e3T^{2} \)
97 \( 1 - 92.5iT - 9.40e3T^{2} \)
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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

−12.36311102727253511143362353786, −11.19423165106490314661662516960, −11.01326640698597079148448206825, −9.600977878859869365320021514517, −8.781072740347580130609859848894, −7.69646749835785141607942670349, −5.63565460334961910824001725169, −4.53187534887190704264862734704, −3.34765658351767904380844184927, −1.45564247889116971052466410428, 1.39812903890127469606548487039, 4.06032883422005079574104313238, 5.28823348075436297884783291731, 6.53884568006532349968954336680, 7.46458072871049998838020961454, 8.548693553465877933699314341913, 9.235492904667826196830457282412, 10.94340145425134662456722869548, 11.87341125775997017939950243358, 13.03405994739032513753669608148

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