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Properties

Label 2-117-117.50-c1-0-8
Degree 22
Conductor 117117
Sign 0.857+0.514i0.857 + 0.514i
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  + (−0.137 − 0.513i)2-s + (1.52 + 0.827i)3-s + (1.48 − 0.858i)4-s + (−0.764 − 2.85i)5-s + (0.215 − 0.894i)6-s + (−2.81 + 2.81i)7-s + (−1.39 − 1.39i)8-s + (1.63 + 2.51i)9-s + (−1.35 + 0.784i)10-s + (1.93 − 0.519i)11-s + (2.97 − 0.0757i)12-s + (−2.59 + 2.50i)13-s + (1.83 + 1.05i)14-s + (1.19 − 4.97i)15-s + (1.19 − 2.06i)16-s + (−2.38 + 4.12i)17-s + ⋯
L(s)  = 1  + (−0.0972 − 0.362i)2-s + (0.878 + 0.477i)3-s + (0.743 − 0.429i)4-s + (−0.341 − 1.27i)5-s + (0.0879 − 0.365i)6-s + (−1.06 + 1.06i)7-s + (−0.493 − 0.493i)8-s + (0.543 + 0.839i)9-s + (−0.429 + 0.248i)10-s + (0.584 − 0.156i)11-s + (0.858 − 0.0218i)12-s + (−0.720 + 0.693i)13-s + (0.490 + 0.283i)14-s + (0.309 − 1.28i)15-s + (0.298 − 0.516i)16-s + (−0.577 + 1.00i)17-s + ⋯

Functional equation

Λ(s)=(117s/2 ΓC(s) L(s)=((0.857+0.514i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.857 + 0.514i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(117s/2 ΓC(s+1/2) L(s)=((0.857+0.514i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.857 + 0.514i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 117117    =    32⋅133^{2} \cdot 13
Sign: 0.857+0.514i0.857 + 0.514i
Analytic conductor: 0.9342490.934249
Root analytic conductor: 0.9665650.966565
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ117(50,⋅)\chi_{117} (50, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 117, ( :1/2), 0.857+0.514i)(2,\ 117,\ (\ :1/2),\ 0.857 + 0.514i)

Particular Values

L(1)L(1) ≈\approx 1.24642−0.344964i1.24642 - 0.344964i
L(12)L(\frac12) ≈\approx 1.24642−0.344964i1.24642 - 0.344964i
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+(−1.52−0.827i)T 1 + (-1.52 - 0.827i)T
13 1+(2.59−2.50i)T 1 + (2.59 - 2.50i)T
good2 1+(0.137+0.513i)T+(−1.73+i)T2 1 + (0.137 + 0.513i)T + (-1.73 + i)T^{2}
5 1+(0.764+2.85i)T+(−4.33+2.5i)T2 1 + (0.764 + 2.85i)T + (-4.33 + 2.5i)T^{2}
7 1+(2.81−2.81i)T−7iT2 1 + (2.81 - 2.81i)T - 7iT^{2}
11 1+(−1.93+0.519i)T+(9.52−5.5i)T2 1 + (-1.93 + 0.519i)T + (9.52 - 5.5i)T^{2}
17 1+(2.38−4.12i)T+(−8.5−14.7i)T2 1 + (2.38 - 4.12i)T + (-8.5 - 14.7i)T^{2}
19 1+(−3.57+0.958i)T+(16.4−9.5i)T2 1 + (-3.57 + 0.958i)T + (16.4 - 9.5i)T^{2}
23 1+0.938T+23T2 1 + 0.938T + 23T^{2}
29 1+(3.82+2.20i)T+(14.5+25.1i)T2 1 + (3.82 + 2.20i)T + (14.5 + 25.1i)T^{2}
31 1+(0.0653−0.0175i)T+(26.8−15.5i)T2 1 + (0.0653 - 0.0175i)T + (26.8 - 15.5i)T^{2}
37 1+(0.00215+0.000576i)T+(32.0+18.5i)T2 1 + (0.00215 + 0.000576i)T + (32.0 + 18.5i)T^{2}
41 1+(5.69−5.69i)T−41iT2 1 + (5.69 - 5.69i)T - 41iT^{2}
43 1+9.98iT−43T2 1 + 9.98iT - 43T^{2}
47 1+(−1.52+5.69i)T+(−40.7−23.5i)T2 1 + (-1.52 + 5.69i)T + (-40.7 - 23.5i)T^{2}
53 1+7.99iT−53T2 1 + 7.99iT - 53T^{2}
59 1+(−1.12+4.21i)T+(−51.0−29.5i)T2 1 + (-1.12 + 4.21i)T + (-51.0 - 29.5i)T^{2}
61 1−12.3T+61T2 1 - 12.3T + 61T^{2}
67 1+(0.773+0.773i)T+67iT2 1 + (0.773 + 0.773i)T + 67iT^{2}
71 1+(−4.06−15.1i)T+(−61.4+35.5i)T2 1 + (-4.06 - 15.1i)T + (-61.4 + 35.5i)T^{2}
73 1+(−0.854+0.854i)T−73iT2 1 + (-0.854 + 0.854i)T - 73iT^{2}
79 1+(0.501+0.868i)T+(−39.5+68.4i)T2 1 + (0.501 + 0.868i)T + (-39.5 + 68.4i)T^{2}
83 1+(−7.70−2.06i)T+(71.8+41.5i)T2 1 + (-7.70 - 2.06i)T + (71.8 + 41.5i)T^{2}
89 1+(0.189−0.708i)T+(−77.0−44.5i)T2 1 + (0.189 - 0.708i)T + (-77.0 - 44.5i)T^{2}
97 1+(−3.74−3.74i)T+97iT2 1 + (-3.74 - 3.74i)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

−13.25012989915545562176968187316, −12.35673745253753918891535642911, −11.55509384819544424753852861016, −9.943954059661916152486204269100, −9.287231929425512324880797742498, −8.470150905628942957713529519930, −6.80677047820002188970576342453, −5.33941681845534210603754722541, −3.71893607440976739381802980234, −2.13410334660338962379432314476, 2.78913912683301661534737580827, 3.59387305837894436687106784115, 6.46982878878285917350396718887, 7.18546716779194736897495606275, 7.65025555328892502316003229937, 9.367052639282205563556006733812, 10.42412310213625408275124041825, 11.61345724726394065071719712833, 12.68944481002571559799705985905, 13.80096114436670297556191737339

Graph of the ZZ-function along the critical line