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Properties

Label 2-117-117.41-c1-0-6
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(41,⋅)\chi_{117} (41, \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

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

Graph of the ZZ-function along the critical line