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Properties

Label 2-117-117.61-c1-0-2
Degree 22
Conductor 117117
Sign 0.679−0.733i0.679 - 0.733i
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  + (0.602 + 1.04i)2-s + (−1.54 − 0.788i)3-s + (0.274 − 0.476i)4-s + (1.89 + 3.27i)5-s + (−0.106 − 2.08i)6-s + 0.300·7-s + 3.07·8-s + (1.75 + 2.43i)9-s + (−2.27 + 3.94i)10-s + (−0.642 − 1.11i)11-s + (−0.799 + 0.517i)12-s + (−3.31 − 1.42i)13-s + (0.180 + 0.313i)14-s + (−0.333 − 6.54i)15-s + (1.29 + 2.25i)16-s + (−2.63 − 4.56i)17-s + ⋯
L(s)  = 1  + (0.425 + 0.737i)2-s + (−0.890 − 0.455i)3-s + (0.137 − 0.238i)4-s + (0.846 + 1.46i)5-s + (−0.0433 − 0.850i)6-s + 0.113·7-s + 1.08·8-s + (0.585 + 0.810i)9-s + (−0.720 + 1.24i)10-s + (−0.193 − 0.335i)11-s + (−0.230 + 0.149i)12-s + (−0.918 − 0.394i)13-s + (0.0483 + 0.0837i)14-s + (−0.0861 − 1.68i)15-s + (0.324 + 0.562i)16-s + (−0.639 − 1.10i)17-s + ⋯

Functional equation

Λ(s)=(117s/2 ΓC(s) L(s)=((0.679−0.733i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.679 - 0.733i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(117s/2 ΓC(s+1/2) L(s)=((0.679−0.733i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.679 - 0.733i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 117117    =    32⋅133^{2} \cdot 13
Sign: 0.679−0.733i0.679 - 0.733i
Analytic conductor: 0.9342490.934249
Root analytic conductor: 0.9665650.966565
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ117(61,⋅)\chi_{117} (61, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 117, ( :1/2), 0.679−0.733i)(2,\ 117,\ (\ :1/2),\ 0.679 - 0.733i)

Particular Values

L(1)L(1) ≈\approx 1.09688+0.479463i1.09688 + 0.479463i
L(12)L(\frac12) ≈\approx 1.09688+0.479463i1.09688 + 0.479463i
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+(1.54+0.788i)T 1 + (1.54 + 0.788i)T
13 1+(3.31+1.42i)T 1 + (3.31 + 1.42i)T
good2 1+(−0.602−1.04i)T+(−1+1.73i)T2 1 + (-0.602 - 1.04i)T + (-1 + 1.73i)T^{2}
5 1+(−1.89−3.27i)T+(−2.5+4.33i)T2 1 + (-1.89 - 3.27i)T + (-2.5 + 4.33i)T^{2}
7 1−0.300T+7T2 1 - 0.300T + 7T^{2}
11 1+(0.642+1.11i)T+(−5.5+9.52i)T2 1 + (0.642 + 1.11i)T + (-5.5 + 9.52i)T^{2}
17 1+(2.63+4.56i)T+(−8.5+14.7i)T2 1 + (2.63 + 4.56i)T + (-8.5 + 14.7i)T^{2}
19 1+(0.829+1.43i)T+(−9.5+16.4i)T2 1 + (0.829 + 1.43i)T + (-9.5 + 16.4i)T^{2}
23 1+3.34T+23T2 1 + 3.34T + 23T^{2}
29 1+(−4.81−8.34i)T+(−14.5+25.1i)T2 1 + (-4.81 - 8.34i)T + (-14.5 + 25.1i)T^{2}
31 1+(3.29+5.71i)T+(−15.5+26.8i)T2 1 + (3.29 + 5.71i)T + (-15.5 + 26.8i)T^{2}
37 1+(−1.97+3.42i)T+(−18.5−32.0i)T2 1 + (-1.97 + 3.42i)T + (-18.5 - 32.0i)T^{2}
41 1−2.86T+41T2 1 - 2.86T + 41T^{2}
43 1+0.0208T+43T2 1 + 0.0208T + 43T^{2}
47 1+(−0.954+1.65i)T+(−23.5−40.7i)T2 1 + (-0.954 + 1.65i)T + (-23.5 - 40.7i)T^{2}
53 1+7.15T+53T2 1 + 7.15T + 53T^{2}
59 1+(4.36−7.56i)T+(−29.5−51.0i)T2 1 + (4.36 - 7.56i)T + (-29.5 - 51.0i)T^{2}
61 1−5.13T+61T2 1 - 5.13T + 61T^{2}
67 1−8.31T+67T2 1 - 8.31T + 67T^{2}
71 1+(−4.64−8.03i)T+(−35.5+61.4i)T2 1 + (-4.64 - 8.03i)T + (-35.5 + 61.4i)T^{2}
73 1+4.69T+73T2 1 + 4.69T + 73T^{2}
79 1+(−6.65+11.5i)T+(−39.5−68.4i)T2 1 + (-6.65 + 11.5i)T + (-39.5 - 68.4i)T^{2}
83 1+(6.45−11.1i)T+(−41.5−71.8i)T2 1 + (6.45 - 11.1i)T + (-41.5 - 71.8i)T^{2}
89 1+(−3.19+5.54i)T+(−44.5−77.0i)T2 1 + (-3.19 + 5.54i)T + (-44.5 - 77.0i)T^{2}
97 1+3.76T+97T2 1 + 3.76T + 97T^{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

−13.92018510801060379968052741099, −12.91411974129902714982462974219, −11.36970532903384492988146875599, −10.69363334835718735991837114827, −9.818079445174680622180795458007, −7.52535346946499569560748206724, −6.82831642156906060147661831081, −5.97158360607860060507307379422, −4.97282338417640281912927747292, −2.36930153080129256176389279579, 1.84337873332427114977916967759, 4.25204181537777910375218626319, 4.98458042714141340039622862644, 6.35288860106198118873228556418, 8.127683429415816687552642823064, 9.544204175497295497816605956212, 10.33664635562639490140654795396, 11.55577485214670713905627475655, 12.51279110101184115222182364572, 12.82407002170517515528318786705

Graph of the ZZ-function along the critical line