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Properties

Label 2-156-4.3-c2-0-9
Degree $2$
Conductor $156$
Sign $0.342 - 0.939i$
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  + (1.96 + 0.348i)2-s + 1.73i·3-s + (3.75 + 1.37i)4-s − 1.51·5-s + (−0.603 + 3.41i)6-s + 10.1i·7-s + (6.92 + 4.00i)8-s − 2.99·9-s + (−2.98 − 0.528i)10-s − 10.8i·11-s + (−2.37 + 6.50i)12-s − 3.60·13-s + (−3.51 + 19.9i)14-s − 2.62i·15-s + (12.2 + 10.3i)16-s + 30.6·17-s + ⋯
L(s)  = 1  + (0.984 + 0.174i)2-s + 0.577i·3-s + (0.939 + 0.342i)4-s − 0.303·5-s + (−0.100 + 0.568i)6-s + 1.44i·7-s + (0.865 + 0.501i)8-s − 0.333·9-s + (−0.298 − 0.0528i)10-s − 0.982i·11-s + (−0.197 + 0.542i)12-s − 0.277·13-s + (−0.251 + 1.42i)14-s − 0.175i·15-s + (0.764 + 0.644i)16-s + 1.80·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.342 - 0.939i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.342 - 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(156\)    =    \(2^{2} \cdot 3 \cdot 13\)
Sign: $0.342 - 0.939i$
Analytic conductor: \(4.25069\)
Root analytic conductor: \(2.06172\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{156} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 156,\ (\ :1),\ 0.342 - 0.939i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.03998 + 1.42704i\)
\(L(\frac12)\) \(\approx\) \(2.03998 + 1.42704i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.96 - 0.348i)T \)
3 \( 1 - 1.73iT \)
13 \( 1 + 3.60T \)
good5 \( 1 + 1.51T + 25T^{2} \)
7 \( 1 - 10.1iT - 49T^{2} \)
11 \( 1 + 10.8iT - 121T^{2} \)
17 \( 1 - 30.6T + 289T^{2} \)
19 \( 1 + 12.4iT - 361T^{2} \)
23 \( 1 + 1.55iT - 529T^{2} \)
29 \( 1 + 20.7T + 841T^{2} \)
31 \( 1 + 40.6iT - 961T^{2} \)
37 \( 1 - 12.4T + 1.36e3T^{2} \)
41 \( 1 - 18.1T + 1.68e3T^{2} \)
43 \( 1 + 38.7iT - 1.84e3T^{2} \)
47 \( 1 + 22.4iT - 2.20e3T^{2} \)
53 \( 1 - 69.1T + 2.80e3T^{2} \)
59 \( 1 + 82.1iT - 3.48e3T^{2} \)
61 \( 1 + 79.3T + 3.72e3T^{2} \)
67 \( 1 - 78.0iT - 4.48e3T^{2} \)
71 \( 1 - 99.7iT - 5.04e3T^{2} \)
73 \( 1 + 130.T + 5.32e3T^{2} \)
79 \( 1 - 144. iT - 6.24e3T^{2} \)
83 \( 1 + 34.5iT - 6.88e3T^{2} \)
89 \( 1 + 90.7T + 7.92e3T^{2} \)
97 \( 1 - 56.1T + 9.40e3T^{2} \)
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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

−12.85178637008812867385939026453, −11.84238458555708331884961681563, −11.35372626308077838987908806378, −9.940185076340142381269089496088, −8.673489992750683559190752431744, −7.61613998669973326922984956697, −5.90029863440853713730635286782, −5.39117745845024576540936292510, −3.81428804758776996072644633229, −2.63189647301437081413262849789, 1.42830741278324025868283330529, 3.37858081549000570340919397444, 4.52529091720269982304617901101, 5.93280827679171286463790818914, 7.37575008363043077657215432345, 7.60422637340294934728546406284, 9.869502585409887531910987005052, 10.61890963916117419971937232454, 11.88616079024899833802418527982, 12.49507323000079237798532632844

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