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Properties

Label 2-156-13.10-c1-0-1
Degree 22
Conductor 156156
Sign 0.964−0.265i0.964 - 0.265i
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 − 1.73i·5-s + (3 + 1.73i)7-s + (−0.499 + 0.866i)9-s + (3 − 1.73i)11-s + (−2.5 + 2.59i)13-s + (1.49 − 0.866i)15-s + (−1.5 + 2.59i)17-s + (−3 − 1.73i)19-s + 3.46i·21-s + (−3 − 5.19i)23-s + 2.00·25-s − 0.999·27-s + (−4.5 − 7.79i)29-s + (3 + 1.73i)33-s + ⋯
L(s)  = 1  + (0.288 + 0.499i)3-s − 0.774i·5-s + (1.13 + 0.654i)7-s + (−0.166 + 0.288i)9-s + (0.904 − 0.522i)11-s + (−0.693 + 0.720i)13-s + (0.387 − 0.223i)15-s + (−0.363 + 0.630i)17-s + (−0.688 − 0.397i)19-s + 0.755i·21-s + (−0.625 − 1.08i)23-s + 0.400·25-s − 0.192·27-s + (−0.835 − 1.44i)29-s + (0.522 + 0.301i)33-s + ⋯

Functional equation

Λ(s)=(156s/2 ΓC(s) L(s)=((0.964−0.265i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.964 - 0.265i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(156s/2 ΓC(s+1/2) L(s)=((0.964−0.265i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.964 - 0.265i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 156156    =    22⋅3⋅132^{2} \cdot 3 \cdot 13
Sign: 0.964−0.265i0.964 - 0.265i
Analytic conductor: 1.245661.24566
Root analytic conductor: 1.116091.11609
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ156(49,⋅)\chi_{156} (49, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 156, ( :1/2), 0.964−0.265i)(2,\ 156,\ (\ :1/2),\ 0.964 - 0.265i)

Particular Values

L(1)L(1) ≈\approx 1.29705+0.174992i1.29705 + 0.174992i
L(12)L(\frac12) ≈\approx 1.29705+0.174992i1.29705 + 0.174992i
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+1.73iT−5T2 1 + 1.73iT - 5T^{2}
7 1+(−3−1.73i)T+(3.5+6.06i)T2 1 + (-3 - 1.73i)T + (3.5 + 6.06i)T^{2}
11 1+(−3+1.73i)T+(5.5−9.52i)T2 1 + (-3 + 1.73i)T + (5.5 - 9.52i)T^{2}
17 1+(1.5−2.59i)T+(−8.5−14.7i)T2 1 + (1.5 - 2.59i)T + (-8.5 - 14.7i)T^{2}
19 1+(3+1.73i)T+(9.5+16.4i)T2 1 + (3 + 1.73i)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+(4.5+7.79i)T+(−14.5+25.1i)T2 1 + (4.5 + 7.79i)T + (-14.5 + 25.1i)T^{2}
31 1−31T2 1 - 31T^{2}
37 1+(4.5−2.59i)T+(18.5−32.0i)T2 1 + (4.5 - 2.59i)T + (18.5 - 32.0i)T^{2}
41 1+(7.5−4.33i)T+(20.5−35.5i)T2 1 + (7.5 - 4.33i)T + (20.5 - 35.5i)T^{2}
43 1+(1−1.73i)T+(−21.5−37.2i)T2 1 + (1 - 1.73i)T + (-21.5 - 37.2i)T^{2}
47 1−3.46iT−47T2 1 - 3.46iT - 47T^{2}
53 1+9T+53T2 1 + 9T + 53T^{2}
59 1+(−12−6.92i)T+(29.5+51.0i)T2 1 + (-12 - 6.92i)T + (29.5 + 51.0i)T^{2}
61 1+(−5.5+9.52i)T+(−30.5−52.8i)T2 1 + (-5.5 + 9.52i)T + (-30.5 - 52.8i)T^{2}
67 1+(−9+5.19i)T+(33.5−58.0i)T2 1 + (-9 + 5.19i)T + (33.5 - 58.0i)T^{2}
71 1+(−9−5.19i)T+(35.5+61.4i)T2 1 + (-9 - 5.19i)T + (35.5 + 61.4i)T^{2}
73 1−5.19iT−73T2 1 - 5.19iT - 73T^{2}
79 1+8T+79T2 1 + 8T + 79T^{2}
83 1+3.46iT−83T2 1 + 3.46iT - 83T^{2}
89 1+(6−3.46i)T+(44.5−77.0i)T2 1 + (6 - 3.46i)T + (44.5 - 77.0i)T^{2}
97 1+(−6−3.46i)T+(48.5+84.0i)T2 1 + (-6 - 3.46i)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

−12.92969730931887237355674837410, −11.86023146607430678276249578473, −11.15827558278724835044081022692, −9.761667962638959940247021542591, −8.711229340482488262986780144509, −8.249799728223798369499347931976, −6.45984664369464484242152134746, −5.02783433508979592011012781687, −4.16420577549207164277272409888, −2.04421677189285670853746386208, 1.87434699414572112069597478785, 3.64827924370899148887378178731, 5.16726744818237835871522606195, 6.86948085074299929749018448082, 7.44773090777286749523056087306, 8.623581491597419451792796482663, 9.962720711660750300781971406164, 10.96526636825649627149395754262, 11.84788915870168825750305157953, 12.91720984623017817836612683453

Graph of the ZZ-function along the critical line