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Properties

Label 2-156-156.35-c1-0-15
Degree $2$
Conductor $156$
Sign $-0.0217 + 0.999i$
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  + (−0.998 − 1.00i)2-s + (1.10 − 1.33i)3-s + (−0.00424 + 1.99i)4-s + 0.171i·5-s + (−2.43 + 0.230i)6-s + (2.51 − 1.45i)7-s + (2.00 − 1.99i)8-s + (−0.568 − 2.94i)9-s + (0.171 − 0.171i)10-s + (0.674 − 1.16i)11-s + (2.66 + 2.21i)12-s + (−3.29 + 1.45i)13-s + (−3.97 − 1.06i)14-s + (0.229 + 0.189i)15-s + (−3.99 − 0.0169i)16-s + (−4.48 + 2.59i)17-s + ⋯
L(s)  = 1  + (−0.706 − 0.707i)2-s + (0.636 − 0.771i)3-s + (−0.00212 + 0.999i)4-s + 0.0767i·5-s + (−0.995 + 0.0940i)6-s + (0.952 − 0.549i)7-s + (0.709 − 0.704i)8-s + (−0.189 − 0.981i)9-s + (0.0543 − 0.0541i)10-s + (0.203 − 0.352i)11-s + (0.769 + 0.638i)12-s + (−0.915 + 0.403i)13-s + (−1.06 − 0.285i)14-s + (0.0591 + 0.0488i)15-s + (−0.999 − 0.00424i)16-s + (−1.08 + 0.628i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0217 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0217 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(156\)    =    \(2^{2} \cdot 3 \cdot 13\)
Sign: $-0.0217 + 0.999i$
Analytic conductor: \(1.24566\)
Root analytic conductor: \(1.11609\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{156} (35, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 156,\ (\ :1/2),\ -0.0217 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.711045 - 0.726696i\)
\(L(\frac12)\) \(\approx\) \(0.711045 - 0.726696i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.998 + 1.00i)T \)
3 \( 1 + (-1.10 + 1.33i)T \)
13 \( 1 + (3.29 - 1.45i)T \)
good5 \( 1 - 0.171iT - 5T^{2} \)
7 \( 1 + (-2.51 + 1.45i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.674 + 1.16i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (4.48 - 2.59i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-4.90 + 2.82i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.40 - 2.44i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.51 - 1.45i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 - 3.62iT - 31T^{2} \)
37 \( 1 + (1.64 - 2.85i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-4.85 - 2.80i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (6.79 - 3.92i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 - 7.05T + 47T^{2} \)
53 \( 1 - 12.1iT - 53T^{2} \)
59 \( 1 + (4.93 + 8.55i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.259 + 0.449i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (9.87 + 5.70i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-5.79 - 10.0i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 1.09T + 73T^{2} \)
79 \( 1 + 5.99iT - 79T^{2} \)
83 \( 1 + 3.19T + 83T^{2} \)
89 \( 1 + (3.87 + 2.23i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (4.06 + 7.03i)T + (-48.5 + 84.0i)T^{2} \)
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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

−12.53991041300973295689976247763, −11.63321980875721288869176391072, −10.79559197529253396422960356029, −9.467130620392789819333719238088, −8.591596885276333170850408978318, −7.62557279482854313431008553299, −6.80240850409743759143303107859, −4.51219891517369610181560473896, −2.93064962862922354484258146061, −1.43676122674522876608637834470, 2.30809502645547933334630248042, 4.59310915044040556100042198092, 5.43023763345894550160046244124, 7.19024865942437812251643744853, 8.163653536150551535725327904386, 9.015873364647614154493313193913, 9.877484880535690882037427976881, 10.86131038381526053452801288650, 11.97447495853208929399715060968, 13.65594513351761414830401157029

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