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Properties

Label 2-156-13.4-c1-0-1
Degree $2$
Conductor $156$
Sign $0.964 + 0.265i$
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  + (0.5 − 0.866i)3-s + 1.73i·5-s + (3 − 1.73i)7-s + (−0.499 − 0.866i)9-s + (3 + 1.73i)11-s + (−2.5 − 2.59i)13-s + (1.49 + 0.866i)15-s + (−1.5 − 2.59i)17-s + (−3 + 1.73i)19-s − 3.46i·21-s + (−3 + 5.19i)23-s + 2.00·25-s − 0.999·27-s + (−4.5 + 7.79i)29-s + (3 − 1.73i)33-s + ⋯
L(s)  = 1  + (0.288 − 0.499i)3-s + 0.774i·5-s + (1.13 − 0.654i)7-s + (−0.166 − 0.288i)9-s + (0.904 + 0.522i)11-s + (−0.693 − 0.720i)13-s + (0.387 + 0.223i)15-s + (−0.363 − 0.630i)17-s + (−0.688 + 0.397i)19-s − 0.755i·21-s + (−0.625 + 1.08i)23-s + 0.400·25-s − 0.192·27-s + (−0.835 + 1.44i)29-s + (0.522 − 0.301i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.964 + 0.265i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.964 + 0.265i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(156\)    =    \(2^{2} \cdot 3 \cdot 13\)
Sign: $0.964 + 0.265i$
Analytic conductor: \(1.24566\)
Root analytic conductor: \(1.11609\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{156} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 156,\ (\ :1/2),\ 0.964 + 0.265i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.29705 - 0.174992i\)
\(L(\frac12)\) \(\approx\) \(1.29705 - 0.174992i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.5 + 0.866i)T \)
13 \( 1 + (2.5 + 2.59i)T \)
good5 \( 1 - 1.73iT - 5T^{2} \)
7 \( 1 + (-3 + 1.73i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-3 - 1.73i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (1.5 + 2.59i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3 - 1.73i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (3 - 5.19i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (4.5 - 7.79i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 + (4.5 + 2.59i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (7.5 + 4.33i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (1 + 1.73i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 3.46iT - 47T^{2} \)
53 \( 1 + 9T + 53T^{2} \)
59 \( 1 + (-12 + 6.92i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5.5 - 9.52i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-9 - 5.19i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-9 + 5.19i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + 5.19iT - 73T^{2} \)
79 \( 1 + 8T + 79T^{2} \)
83 \( 1 - 3.46iT - 83T^{2} \)
89 \( 1 + (6 + 3.46i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-6 + 3.46i)T + (48.5 - 84.0i)T^{2} \)
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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

−12.91720984623017817836612683453, −11.84788915870168825750305157953, −10.96526636825649627149395754262, −9.962720711660750300781971406164, −8.623581491597419451792796482663, −7.44773090777286749523056087306, −6.86948085074299929749018448082, −5.16726744818237835871522606195, −3.64827924370899148887378178731, −1.87434699414572112069597478785, 2.04421677189285670853746386208, 4.16420577549207164277272409888, 5.02783433508979592011012781687, 6.45984664369464484242152134746, 8.249799728223798369499347931976, 8.711229340482488262986780144509, 9.761667962638959940247021542591, 11.15827558278724835044081022692, 11.86023146607430678276249578473, 12.92969730931887237355674837410

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