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Properties

Label 2-156-13.3-c1-0-0
Degree 22
Conductor 156156
Sign 0.872−0.488i0.872 - 0.488i
Analytic cond. 1.245661.24566
Root an. cond. 1.116091.11609
Motivic weight 11
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 + 2·5-s + (−0.5 + 0.866i)7-s + (−0.499 + 0.866i)9-s + (−1 − 1.73i)11-s + (2.5 + 2.59i)13-s + (1 + 1.73i)15-s + (2 − 3.46i)17-s + (−2 + 3.46i)19-s − 0.999·21-s + (−3 − 5.19i)23-s − 25-s − 0.999·27-s + (−3 − 5.19i)29-s − 31-s + ⋯
L(s)  = 1  + (0.288 + 0.499i)3-s + 0.894·5-s + (−0.188 + 0.327i)7-s + (−0.166 + 0.288i)9-s + (−0.301 − 0.522i)11-s + (0.693 + 0.720i)13-s + (0.258 + 0.447i)15-s + (0.485 − 0.840i)17-s + (−0.458 + 0.794i)19-s − 0.218·21-s + (−0.625 − 1.08i)23-s − 0.200·25-s − 0.192·27-s + (−0.557 − 0.964i)29-s − 0.179·31-s + ⋯

Functional equation

Λ(s)=(156s/2 ΓC(s) L(s)=((0.872−0.488i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.872 - 0.488i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(156s/2 ΓC(s+1/2) L(s)=((0.872−0.488i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.872 - 0.488i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 156156    =    22⋅3⋅132^{2} \cdot 3 \cdot 13
Sign: 0.872−0.488i0.872 - 0.488i
Analytic conductor: 1.245661.24566
Root analytic conductor: 1.116091.11609
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ156(133,⋅)\chi_{156} (133, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 156, ( :1/2), 0.872−0.488i)(2,\ 156,\ (\ :1/2),\ 0.872 - 0.488i)

Particular Values

L(1)L(1) ≈\approx 1.28992+0.336785i1.28992 + 0.336785i
L(12)L(\frac12) ≈\approx 1.28992+0.336785i1.28992 + 0.336785i
L(32)L(\frac{3}{2}) 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+(−2.5−2.59i)T 1 + (-2.5 - 2.59i)T
good5 1−2T+5T2 1 - 2T + 5T^{2}
7 1+(0.5−0.866i)T+(−3.5−6.06i)T2 1 + (0.5 - 0.866i)T + (-3.5 - 6.06i)T^{2}
11 1+(1+1.73i)T+(−5.5+9.52i)T2 1 + (1 + 1.73i)T + (-5.5 + 9.52i)T^{2}
17 1+(−2+3.46i)T+(−8.5−14.7i)T2 1 + (-2 + 3.46i)T + (-8.5 - 14.7i)T^{2}
19 1+(2−3.46i)T+(−9.5−16.4i)T2 1 + (2 - 3.46i)T + (-9.5 - 16.4i)T^{2}
23 1+(3+5.19i)T+(−11.5+19.9i)T2 1 + (3 + 5.19i)T + (-11.5 + 19.9i)T^{2}
29 1+(3+5.19i)T+(−14.5+25.1i)T2 1 + (3 + 5.19i)T + (-14.5 + 25.1i)T^{2}
31 1+T+31T2 1 + T + 31T^{2}
37 1+(5+8.66i)T+(−18.5+32.0i)T2 1 + (5 + 8.66i)T + (-18.5 + 32.0i)T^{2}
41 1+(−2−3.46i)T+(−20.5+35.5i)T2 1 + (-2 - 3.46i)T + (-20.5 + 35.5i)T^{2}
43 1+(0.5−0.866i)T+(−21.5−37.2i)T2 1 + (0.5 - 0.866i)T + (-21.5 - 37.2i)T^{2}
47 1+10T+47T2 1 + 10T + 47T^{2}
53 1−8T+53T2 1 - 8T + 53T^{2}
59 1+(−1+1.73i)T+(−29.5−51.0i)T2 1 + (-1 + 1.73i)T + (-29.5 - 51.0i)T^{2}
61 1+(−2.5+4.33i)T+(−30.5−52.8i)T2 1 + (-2.5 + 4.33i)T + (-30.5 - 52.8i)T^{2}
67 1+(−3.5−6.06i)T+(−33.5+58.0i)T2 1 + (-3.5 - 6.06i)T + (-33.5 + 58.0i)T^{2}
71 1+(5−8.66i)T+(−35.5−61.4i)T2 1 + (5 - 8.66i)T + (-35.5 - 61.4i)T^{2}
73 1+7T+73T2 1 + 7T + 73T^{2}
79 1−17T+79T2 1 - 17T + 79T^{2}
83 1−12T+83T2 1 - 12T + 83T^{2}
89 1+(−8−13.8i)T+(−44.5+77.0i)T2 1 + (-8 - 13.8i)T + (-44.5 + 77.0i)T^{2}
97 1+(6.5−11.2i)T+(−48.5−84.0i)T2 1 + (6.5 - 11.2i)T + (-48.5 - 84.0i)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

−13.22929405833091896234376980300, −12.00842269388811516573612185027, −10.85887723372832577648194829179, −9.882345580425580149407445116653, −9.084440555132997721424777055273, −8.020467363895559666625534347247, −6.37012680269715102469464767428, −5.46363969577617734799272243899, −3.88926409756123921838097094583, −2.28208696086453269741944568752, 1.79363522398815726296353600104, 3.49047385926898507688183167604, 5.36033811416728972105281609698, 6.42820135966026690877177030048, 7.60133923080677457889925424303, 8.712898833827552243344155258594, 9.875127469912000617867265813772, 10.67066691721290065628693802890, 12.04717154581598833067683578174, 13.22413003857927670446620885563

Graph of the ZZ-function along the critical line