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Properties

Label 2-156-52.11-c1-0-8
Degree $2$
Conductor $156$
Sign $-0.438 + 0.898i$
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  + (−1.41 − 0.0898i)2-s + (−0.866 + 0.5i)3-s + (1.98 + 0.253i)4-s + (−1.54 − 1.54i)5-s + (1.26 − 0.627i)6-s + (−1.07 + 0.286i)7-s + (−2.77 − 0.536i)8-s + (0.499 − 0.866i)9-s + (2.04 + 2.31i)10-s + (1.02 − 3.82i)11-s + (−1.84 + 0.772i)12-s + (−2.27 − 2.80i)13-s + (1.53 − 0.308i)14-s + (2.11 + 0.565i)15-s + (3.87 + 1.00i)16-s + (−5.43 − 3.13i)17-s + ⋯
L(s)  = 1  + (−0.997 − 0.0635i)2-s + (−0.499 + 0.288i)3-s + (0.991 + 0.126i)4-s + (−0.690 − 0.690i)5-s + (0.517 − 0.256i)6-s + (−0.404 + 0.108i)7-s + (−0.981 − 0.189i)8-s + (0.166 − 0.288i)9-s + (0.645 + 0.733i)10-s + (0.308 − 1.15i)11-s + (−0.532 + 0.222i)12-s + (−0.629 − 0.776i)13-s + (0.410 − 0.0824i)14-s + (0.544 + 0.145i)15-s + (0.967 + 0.251i)16-s + (−1.31 − 0.760i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.438 + 0.898i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.438 + 0.898i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(156\)    =    \(2^{2} \cdot 3 \cdot 13\)
Sign: $-0.438 + 0.898i$
Analytic conductor: \(1.24566\)
Root analytic conductor: \(1.11609\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{156} (115, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 156,\ (\ :1/2),\ -0.438 + 0.898i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.184741 - 0.295836i\)
\(L(\frac12)\) \(\approx\) \(0.184741 - 0.295836i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.41 + 0.0898i)T \)
3 \( 1 + (0.866 - 0.5i)T \)
13 \( 1 + (2.27 + 2.80i)T \)
good5 \( 1 + (1.54 + 1.54i)T + 5iT^{2} \)
7 \( 1 + (1.07 - 0.286i)T + (6.06 - 3.5i)T^{2} \)
11 \( 1 + (-1.02 + 3.82i)T + (-9.52 - 5.5i)T^{2} \)
17 \( 1 + (5.43 + 3.13i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.36 + 5.08i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (-2.78 - 4.82i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.42 - 4.20i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (3.61 - 3.61i)T - 31iT^{2} \)
37 \( 1 + (-6.94 - 1.86i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (-0.144 + 0.539i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (-1.50 + 2.60i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-1.42 - 1.42i)T + 47iT^{2} \)
53 \( 1 - 0.361T + 53T^{2} \)
59 \( 1 + (5.88 - 1.57i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (-4.29 + 7.43i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (10.8 + 2.89i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (0.111 + 0.414i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (-1.32 + 1.32i)T - 73iT^{2} \)
79 \( 1 + 9.25iT - 79T^{2} \)
83 \( 1 + (-12.6 + 12.6i)T - 83iT^{2} \)
89 \( 1 + (-15.3 - 4.11i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (6.77 - 1.81i)T + (84.0 - 48.5i)T^{2} \)
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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

−12.30648178339568474446289128478, −11.37789453276965543552005228220, −10.74322812117437335800956627975, −9.320318070916043931064220607344, −8.754363098684610020395981773395, −7.47889260676381350688766087003, −6.33816894525102992600081758757, −4.90002169453943761706033039589, −3.10765058082920156098705189265, −0.46933631867377373141816619696, 2.20143233561997997657490919370, 4.21683340358179902131927179067, 6.30846781405721995590933283422, 6.97436806058876141080502985510, 7.903529452809083550004469150426, 9.273095819630731368411380013940, 10.28095988220395612401263404072, 11.15366150261919442496823865549, 12.01307200061032001101119806725, 12.88076610758175894241736985851

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