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Properties

Label 2-117-117.20-c1-0-0
Degree 22
Conductor 117117
Sign −0.0602−0.998i-0.0602 - 0.998i
Analytic cond. 0.9342490.934249
Root an. cond. 0.9665650.966565
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  + (−1.02 − 0.275i)2-s + (−1.73 + 0.00536i)3-s + (−0.747 − 0.431i)4-s + (2.22 + 0.595i)5-s + (1.78 + 0.472i)6-s + (−3.66 + 3.66i)7-s + (2.15 + 2.15i)8-s + (2.99 − 0.0185i)9-s + (−2.12 − 1.22i)10-s + (−0.708 + 2.64i)11-s + (1.29 + 0.743i)12-s + (−2.69 + 2.39i)13-s + (4.78 − 2.76i)14-s + (−3.85 − 1.01i)15-s + (−0.764 − 1.32i)16-s + (−0.0716 − 0.124i)17-s + ⋯
L(s)  = 1  + (−0.728 − 0.195i)2-s + (−0.999 + 0.00309i)3-s + (−0.373 − 0.215i)4-s + (0.993 + 0.266i)5-s + (0.728 + 0.192i)6-s + (−1.38 + 1.38i)7-s + (0.763 + 0.763i)8-s + (0.999 − 0.00619i)9-s + (−0.671 − 0.387i)10-s + (−0.213 + 0.797i)11-s + (0.374 + 0.214i)12-s + (−0.747 + 0.664i)13-s + (1.27 − 0.737i)14-s + (−0.994 − 0.263i)15-s + (−0.191 − 0.331i)16-s + (−0.0173 − 0.0300i)17-s + ⋯

Functional equation

Λ(s)=(117s/2 ΓC(s) L(s)=((−0.0602−0.998i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0602 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(117s/2 ΓC(s+1/2) L(s)=((−0.0602−0.998i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0602 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 117117    =    32⋅133^{2} \cdot 13
Sign: −0.0602−0.998i-0.0602 - 0.998i
Analytic conductor: 0.9342490.934249
Root analytic conductor: 0.9665650.966565
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ117(20,⋅)\chi_{117} (20, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 117, ( :1/2), −0.0602−0.998i)(2,\ 117,\ (\ :1/2),\ -0.0602 - 0.998i)

Particular Values

L(1)L(1) ≈\approx 0.255758+0.271668i0.255758 + 0.271668i
L(12)L(\frac12) ≈\approx 0.255758+0.271668i0.255758 + 0.271668i
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)
bad3 1+(1.73−0.00536i)T 1 + (1.73 - 0.00536i)T
13 1+(2.69−2.39i)T 1 + (2.69 - 2.39i)T
good2 1+(1.02+0.275i)T+(1.73+i)T2 1 + (1.02 + 0.275i)T + (1.73 + i)T^{2}
5 1+(−2.22−0.595i)T+(4.33+2.5i)T2 1 + (-2.22 - 0.595i)T + (4.33 + 2.5i)T^{2}
7 1+(3.66−3.66i)T−7iT2 1 + (3.66 - 3.66i)T - 7iT^{2}
11 1+(0.708−2.64i)T+(−9.52−5.5i)T2 1 + (0.708 - 2.64i)T + (-9.52 - 5.5i)T^{2}
17 1+(0.0716+0.124i)T+(−8.5+14.7i)T2 1 + (0.0716 + 0.124i)T + (-8.5 + 14.7i)T^{2}
19 1+(−0.552+2.06i)T+(−16.4−9.5i)T2 1 + (-0.552 + 2.06i)T + (-16.4 - 9.5i)T^{2}
23 1−0.843T+23T2 1 - 0.843T + 23T^{2}
29 1+(−0.523+0.302i)T+(14.5−25.1i)T2 1 + (-0.523 + 0.302i)T + (14.5 - 25.1i)T^{2}
31 1+(0.905−3.38i)T+(−26.8−15.5i)T2 1 + (0.905 - 3.38i)T + (-26.8 - 15.5i)T^{2}
37 1+(−0.973−3.63i)T+(−32.0+18.5i)T2 1 + (-0.973 - 3.63i)T + (-32.0 + 18.5i)T^{2}
41 1+(5.53−5.53i)T−41iT2 1 + (5.53 - 5.53i)T - 41iT^{2}
43 1+0.216iT−43T2 1 + 0.216iT - 43T^{2}
47 1+(−10.1+2.72i)T+(40.7−23.5i)T2 1 + (-10.1 + 2.72i)T + (40.7 - 23.5i)T^{2}
53 1−8.00iT−53T2 1 - 8.00iT - 53T^{2}
59 1+(−7.19+1.92i)T+(51.0−29.5i)T2 1 + (-7.19 + 1.92i)T + (51.0 - 29.5i)T^{2}
61 1+1.35T+61T2 1 + 1.35T + 61T^{2}
67 1+(−5.41−5.41i)T+67iT2 1 + (-5.41 - 5.41i)T + 67iT^{2}
71 1+(−7.04−1.88i)T+(61.4+35.5i)T2 1 + (-7.04 - 1.88i)T + (61.4 + 35.5i)T^{2}
73 1+(3.76−3.76i)T−73iT2 1 + (3.76 - 3.76i)T - 73iT^{2}
79 1+(−1.62+2.81i)T+(−39.5−68.4i)T2 1 + (-1.62 + 2.81i)T + (-39.5 - 68.4i)T^{2}
83 1+(3.94+14.7i)T+(−71.8+41.5i)T2 1 + (3.94 + 14.7i)T + (-71.8 + 41.5i)T^{2}
89 1+(−14.2+3.81i)T+(77.0−44.5i)T2 1 + (-14.2 + 3.81i)T + (77.0 - 44.5i)T^{2}
97 1+(0.384+0.384i)T+97iT2 1 + (0.384 + 0.384i)T + 97iT^{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

−13.55179374779226921958271347477, −12.64803616129900607347726563857, −11.71457804718099417373654284329, −10.22544134324124806882411040354, −9.756852369191603898379801518154, −8.992886105577169873852770027172, −6.99027441796383237487896515636, −5.93906176054755214611274240389, −4.93954795101334370904259628374, −2.23753387386426308761858275196, 0.57955395697280556600847055186, 3.83038790592677598009065727992, 5.46243675510767464930568099660, 6.66741994761423500589561260182, 7.67119790295221443841111032024, 9.376464946810177573490934186101, 10.05252772650841819394016288526, 10.68136639686878911858834803366, 12.51381850182485642913915219396, 13.20608429269621993405266921701

Graph of the ZZ-function along the critical line