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Properties

Label 2-156-39.29-c0-0-0
Degree 22
Conductor 156156
Sign 0.711+0.702i0.711 + 0.702i
Analytic cond. 0.07785410.0778541
Root an. cond. 0.2790230.279023
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 − 0.866i)3-s + (0.5 − 0.866i)7-s + (−0.499 + 0.866i)9-s + (−0.5 + 0.866i)13-s + (−1 + 1.73i)19-s − 0.999·21-s + 25-s + 0.999·27-s − 31-s + (−1 − 1.73i)37-s + 0.999·39-s + (0.5 − 0.866i)43-s + 1.99·57-s + (0.5 − 0.866i)61-s + (0.499 + 0.866i)63-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)3-s + (0.5 − 0.866i)7-s + (−0.499 + 0.866i)9-s + (−0.5 + 0.866i)13-s + (−1 + 1.73i)19-s − 0.999·21-s + 25-s + 0.999·27-s − 31-s + (−1 − 1.73i)37-s + 0.999·39-s + (0.5 − 0.866i)43-s + 1.99·57-s + (0.5 − 0.866i)61-s + (0.499 + 0.866i)63-s + ⋯

Functional equation

Λ(s)=(156s/2 ΓC(s) L(s)=((0.711+0.702i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.711 + 0.702i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(156s/2 ΓC(s) L(s)=((0.711+0.702i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.711 + 0.702i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 156156    =    22⋅3⋅132^{2} \cdot 3 \cdot 13
Sign: 0.711+0.702i0.711 + 0.702i
Analytic conductor: 0.07785410.0778541
Root analytic conductor: 0.2790230.279023
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ156(29,⋅)\chi_{156} (29, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 156, ( :0), 0.711+0.702i)(2,\ 156,\ (\ :0),\ 0.711 + 0.702i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.57488673280.5748867328
L(12)L(\frac12) ≈\approx 0.57488673280.5748867328
L(1)L(1) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1 1
3 1+(0.5+0.866i)T 1 + (0.5 + 0.866i)T
13 1+(0.5−0.866i)T 1 + (0.5 - 0.866i)T
good5 1−T2 1 - T^{2}
7 1+(−0.5+0.866i)T+(−0.5−0.866i)T2 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2}
11 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
17 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
19 1+(1−1.73i)T+(−0.5−0.866i)T2 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2}
23 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
29 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
31 1+T+T2 1 + T + T^{2}
37 1+(1+1.73i)T+(−0.5+0.866i)T2 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2}
41 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
43 1+(−0.5+0.866i)T+(−0.5−0.866i)T2 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2}
47 1−T2 1 - T^{2}
53 1−T2 1 - T^{2}
59 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
61 1+(−0.5+0.866i)T+(−0.5−0.866i)T2 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2}
67 1+(−0.5−0.866i)T+(−0.5+0.866i)T2 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2}
71 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
73 1+T+T2 1 + T + T^{2}
79 1+T+T2 1 + T + T^{2}
83 1−T2 1 - T^{2}
89 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
97 1+(−0.5+0.866i)T+(−0.5−0.866i)T2 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(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

−12.87467162212949865026756229537, −12.19828706702413409649508990299, −11.07231596751651044830586872506, −10.34906402372767019976987561544, −8.761415698778100303896616633042, −7.61222166988859049414115298574, −6.82635072436709119164717233769, −5.53519951937327730915506033297, −4.12196296708099721647063286021, −1.85492671042857562741002517517, 2.86001860257039679398210868800, 4.64419865390129387614011547920, 5.45461149728077424722847653941, 6.78205128267635501739595218871, 8.426320178916794271269645742399, 9.223432634792648971063271686512, 10.42865324778073088288378406500, 11.21618870351214128473596623863, 12.16223898308493711616487105926, 13.13712256832366131695424890221

Graph of the ZZ-function along the critical line