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Properties

Label 2-156-52.15-c1-0-11
Degree $2$
Conductor $156$
Sign $-0.175 + 0.984i$
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  + (−0.101 − 1.41i)2-s + (0.866 − 0.5i)3-s + (−1.97 + 0.287i)4-s + (2.43 − 2.43i)5-s + (−0.793 − 1.17i)6-s + (0.522 + 1.95i)7-s + (0.607 + 2.76i)8-s + (0.499 − 0.866i)9-s + (−3.67 − 3.18i)10-s + (−3.03 − 0.812i)11-s + (−1.57 + 1.23i)12-s + (−3.44 − 1.04i)13-s + (2.69 − 0.936i)14-s + (0.890 − 3.32i)15-s + (3.83 − 1.13i)16-s + (4.36 + 2.52i)17-s + ⋯
L(s)  = 1  + (−0.0720 − 0.997i)2-s + (0.499 − 0.288i)3-s + (−0.989 + 0.143i)4-s + (1.08 − 1.08i)5-s + (−0.323 − 0.477i)6-s + (0.197 + 0.737i)7-s + (0.214 + 0.976i)8-s + (0.166 − 0.288i)9-s + (−1.16 − 1.00i)10-s + (−0.914 − 0.245i)11-s + (−0.453 + 0.357i)12-s + (−0.956 − 0.291i)13-s + (0.721 − 0.250i)14-s + (0.229 − 0.857i)15-s + (0.958 − 0.284i)16-s + (1.05 + 0.611i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.175 + 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.175 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(156\)    =    \(2^{2} \cdot 3 \cdot 13\)
Sign: $-0.175 + 0.984i$
Analytic conductor: \(1.24566\)
Root analytic conductor: \(1.11609\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{156} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 156,\ (\ :1/2),\ -0.175 + 0.984i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.826296 - 0.987069i\)
\(L(\frac12)\) \(\approx\) \(0.826296 - 0.987069i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.101 + 1.41i)T \)
3 \( 1 + (-0.866 + 0.5i)T \)
13 \( 1 + (3.44 + 1.04i)T \)
good5 \( 1 + (-2.43 + 2.43i)T - 5iT^{2} \)
7 \( 1 + (-0.522 - 1.95i)T + (-6.06 + 3.5i)T^{2} \)
11 \( 1 + (3.03 + 0.812i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (-4.36 - 2.52i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.75 - 0.737i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (-2.20 - 3.82i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.754 - 1.30i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-2.38 - 2.38i)T + 31iT^{2} \)
37 \( 1 + (1.91 - 7.13i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (-3.46 - 0.929i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (2.78 - 4.82i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-8.53 + 8.53i)T - 47iT^{2} \)
53 \( 1 + 9.42T + 53T^{2} \)
59 \( 1 + (2.67 + 9.97i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (5.34 - 9.25i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.40 + 12.7i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (10.4 - 2.80i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (5.57 + 5.57i)T + 73iT^{2} \)
79 \( 1 - 11.8iT - 79T^{2} \)
83 \( 1 + (8.12 + 8.12i)T + 83iT^{2} \)
89 \( 1 + (0.935 - 3.49i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (1.02 + 3.80i)T + (-84.0 + 48.5i)T^{2} \)
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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

−12.67470853002791275308002898105, −12.00767151190237697801707570593, −10.42055426847396796119502493111, −9.647036000319234190505891176059, −8.729935972011657533297503717012, −7.938751540093141478705734043594, −5.69907553039484328412089481115, −4.89608784740623463865435994103, −2.89861073075039954961427237744, −1.63048018222771508603542018188, 2.69292486471460217673710920690, 4.50080947686537653106349946105, 5.75654303661959461259433268988, 7.05437619895058688264108142463, 7.68756873698201073567055875626, 9.189942705295902488121867741230, 10.11040313652170317883659132467, 10.62349098744635421256280229614, 12.64374612565551311191913275628, 13.71837206630682063056489734678

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