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Properties

Label 2-156-52.7-c1-0-5
Degree $2$
Conductor $156$
Sign $0.0175 - 0.999i$
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  + (0.806 + 1.16i)2-s + (0.866 + 0.5i)3-s + (−0.700 + 1.87i)4-s + (−0.922 − 0.922i)5-s + (0.117 + 1.40i)6-s + (−0.757 + 2.82i)7-s + (−2.74 + 0.696i)8-s + (0.499 + 0.866i)9-s + (0.328 − 1.81i)10-s + (6.23 − 1.67i)11-s + (−1.54 + 1.27i)12-s + (−1.11 − 3.42i)13-s + (−3.89 + 1.39i)14-s + (−0.337 − 1.26i)15-s + (−3.01 − 2.62i)16-s + (4.26 − 2.46i)17-s + ⋯
L(s)  = 1  + (0.569 + 0.821i)2-s + (0.499 + 0.288i)3-s + (−0.350 + 0.936i)4-s + (−0.412 − 0.412i)5-s + (0.0478 + 0.575i)6-s + (−0.286 + 1.06i)7-s + (−0.969 + 0.246i)8-s + (0.166 + 0.288i)9-s + (0.103 − 0.574i)10-s + (1.88 − 0.504i)11-s + (−0.445 + 0.367i)12-s + (−0.309 − 0.950i)13-s + (−1.04 + 0.373i)14-s + (−0.0872 − 0.325i)15-s + (−0.754 − 0.656i)16-s + (1.03 − 0.597i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0175 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0175 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(156\)    =    \(2^{2} \cdot 3 \cdot 13\)
Sign: $0.0175 - 0.999i$
Analytic conductor: \(1.24566\)
Root analytic conductor: \(1.11609\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{156} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 156,\ (\ :1/2),\ 0.0175 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.10606 + 1.08677i\)
\(L(\frac12)\) \(\approx\) \(1.10606 + 1.08677i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.806 - 1.16i)T \)
3 \( 1 + (-0.866 - 0.5i)T \)
13 \( 1 + (1.11 + 3.42i)T \)
good5 \( 1 + (0.922 + 0.922i)T + 5iT^{2} \)
7 \( 1 + (0.757 - 2.82i)T + (-6.06 - 3.5i)T^{2} \)
11 \( 1 + (-6.23 + 1.67i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 + (-4.26 + 2.46i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (5.87 + 1.57i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (1.59 - 2.76i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.83 + 3.17i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (1.77 - 1.77i)T - 31iT^{2} \)
37 \( 1 + (-1.65 - 6.17i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + (8.16 - 2.18i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (-0.803 - 1.39i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (1.94 + 1.94i)T + 47iT^{2} \)
53 \( 1 - 2.19T + 53T^{2} \)
59 \( 1 + (-0.998 + 3.72i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (-1.35 - 2.35i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.55 - 5.80i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (6.51 + 1.74i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 + (7.04 - 7.04i)T - 73iT^{2} \)
79 \( 1 - 3.26iT - 79T^{2} \)
83 \( 1 + (1.56 - 1.56i)T - 83iT^{2} \)
89 \( 1 + (2.53 + 9.46i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (1.76 - 6.59i)T + (-84.0 - 48.5i)T^{2} \)
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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

−13.30581656090132846560465948257, −12.23028647737126318943570648017, −11.71843899224494861943201981780, −9.762492072921853755938344643846, −8.763402333659254677154503855138, −8.162649234843220464591237870456, −6.67578300915422323134267529692, −5.59198720797124907069990876675, −4.26000665903644064148167470268, −3.05695493553483011619238011783, 1.67635707011694025500925682234, 3.65681015801774233509223706860, 4.21279934659059444244537417655, 6.36955839445900720125847697918, 7.16649554097056479511995821490, 8.819887524186037127503866618165, 9.828118330048824361140818336874, 10.76577094859876928297734336711, 11.89684367653380870820071511365, 12.55945453011186421401938570436

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