Properties

Label 2-156-52.51-c2-0-1
Degree $2$
Conductor $156$
Sign $-0.687 + 0.725i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.687 + 0.725i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.687 + 0.725i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(156\)    =    \(2^{2} \cdot 3 \cdot 13\)
Sign: $-0.687 + 0.725i$
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: \((2,\ 156,\ (\ :1),\ -0.687 + 0.725i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.184200 - 0.428394i\)
\(L(\frac12)\) \(\approx\) \(0.184200 - 0.428394i\)
\(L(2)\) 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

−13.18894727101673652905125742958, −12.91870758452909702859605942351, −11.17684274496300210625660709427, −9.952990408533650566397347699154, −9.228108932805432700901888691659, −8.180322175579982634268490133802, −6.80618180329251864135490228254, −5.91432779039612411444062958186, −4.60216726488220821148779961367, −3.31296632944344550688518071678, 0.27959376650461645845677153908, 2.44677179757561465173463141649, 3.59862903057064178601520946888, 5.37570536373488053617839155450, 6.60471024632067642812132719559, 8.104075502767329772065024431850, 9.100100973968937774804558984780, 10.32197836001509962770694807798, 10.93211173901133260767558819447, 12.34703294709883989611892411852

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