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Properties

Label 2-117-117.43-c1-0-1
Degree 22
Conductor 117117
Sign 0.0469−0.998i0.0469 - 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  + (0.495 − 0.285i)2-s + (−1.06 + 1.36i)3-s + (−0.836 + 1.44i)4-s + (−0.796 + 0.459i)5-s + (−0.133 + 0.981i)6-s + 1.93i·7-s + 2.10i·8-s + (−0.751 − 2.90i)9-s + (−0.262 + 0.455i)10-s + (3.64 − 2.10i)11-s + (−1.09 − 2.68i)12-s + (1.35 + 3.34i)13-s + (0.552 + 0.957i)14-s + (0.214 − 1.57i)15-s + (−1.07 − 1.85i)16-s + (−1.20 − 2.08i)17-s + ⋯
L(s)  = 1  + (0.350 − 0.202i)2-s + (−0.612 + 0.790i)3-s + (−0.418 + 0.724i)4-s + (−0.356 + 0.205i)5-s + (−0.0544 + 0.400i)6-s + 0.730i·7-s + 0.742i·8-s + (−0.250 − 0.968i)9-s + (−0.0831 + 0.143i)10-s + (1.09 − 0.633i)11-s + (−0.316 − 0.774i)12-s + (0.375 + 0.926i)13-s + (0.147 + 0.255i)14-s + (0.0553 − 0.407i)15-s + (−0.268 − 0.464i)16-s + (−0.291 − 0.505i)17-s + ⋯

Functional equation

Λ(s)=(117s/2 ΓC(s) L(s)=((0.0469−0.998i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0469 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(117s/2 ΓC(s+1/2) L(s)=((0.0469−0.998i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0469 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 117117    =    32⋅133^{2} \cdot 13
Sign: 0.0469−0.998i0.0469 - 0.998i
Analytic conductor: 0.9342490.934249
Root analytic conductor: 0.9665650.966565
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ117(43,⋅)\chi_{117} (43, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 117, ( :1/2), 0.0469−0.998i)(2,\ 117,\ (\ :1/2),\ 0.0469 - 0.998i)

Particular Values

L(1)L(1) ≈\approx 0.656569+0.626463i0.656569 + 0.626463i
L(12)L(\frac12) ≈\approx 0.656569+0.626463i0.656569 + 0.626463i
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+(1.06−1.36i)T 1 + (1.06 - 1.36i)T
13 1+(−1.35−3.34i)T 1 + (-1.35 - 3.34i)T
good2 1+(−0.495+0.285i)T+(1−1.73i)T2 1 + (-0.495 + 0.285i)T + (1 - 1.73i)T^{2}
5 1+(0.796−0.459i)T+(2.5−4.33i)T2 1 + (0.796 - 0.459i)T + (2.5 - 4.33i)T^{2}
7 1−1.93iT−7T2 1 - 1.93iT - 7T^{2}
11 1+(−3.64+2.10i)T+(5.5−9.52i)T2 1 + (-3.64 + 2.10i)T + (5.5 - 9.52i)T^{2}
17 1+(1.20+2.08i)T+(−8.5+14.7i)T2 1 + (1.20 + 2.08i)T + (-8.5 + 14.7i)T^{2}
19 1+(−1.60+0.928i)T+(9.5−16.4i)T2 1 + (-1.60 + 0.928i)T + (9.5 - 16.4i)T^{2}
23 1−8.22T+23T2 1 - 8.22T + 23T^{2}
29 1+(2.36+4.09i)T+(−14.5+25.1i)T2 1 + (2.36 + 4.09i)T + (-14.5 + 25.1i)T^{2}
31 1+(4.29−2.47i)T+(15.5−26.8i)T2 1 + (4.29 - 2.47i)T + (15.5 - 26.8i)T^{2}
37 1+(0.959+0.554i)T+(18.5+32.0i)T2 1 + (0.959 + 0.554i)T + (18.5 + 32.0i)T^{2}
41 1+0.566iT−41T2 1 + 0.566iT - 41T^{2}
43 1−9.58T+43T2 1 - 9.58T + 43T^{2}
47 1+(1.35+0.780i)T+(23.5+40.7i)T2 1 + (1.35 + 0.780i)T + (23.5 + 40.7i)T^{2}
53 1+4.09T+53T2 1 + 4.09T + 53T^{2}
59 1+(−5.29−3.05i)T+(29.5+51.0i)T2 1 + (-5.29 - 3.05i)T + (29.5 + 51.0i)T^{2}
61 1−1.33T+61T2 1 - 1.33T + 61T^{2}
67 1+16.3iT−67T2 1 + 16.3iT - 67T^{2}
71 1+(10.6−6.12i)T+(35.5−61.4i)T2 1 + (10.6 - 6.12i)T + (35.5 - 61.4i)T^{2}
73 1−8.77iT−73T2 1 - 8.77iT - 73T^{2}
79 1+(−4.09+7.09i)T+(−39.5−68.4i)T2 1 + (-4.09 + 7.09i)T + (-39.5 - 68.4i)T^{2}
83 1+(2.31+1.33i)T+(41.5+71.8i)T2 1 + (2.31 + 1.33i)T + (41.5 + 71.8i)T^{2}
89 1+(11.4+6.61i)T+(44.5+77.0i)T2 1 + (11.4 + 6.61i)T + (44.5 + 77.0i)T^{2}
97 1+13.8iT−97T2 1 + 13.8iT - 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

−13.79709994179759040359933286242, −12.53648760097026639997745498018, −11.43855915762074736893149844158, −11.30286080502540533400108384054, −9.236011373954407537924878796611, −8.901218308659970390179310945596, −7.06474143637873597026215420957, −5.62519647458676794965417894053, −4.33924042196526366954104719069, −3.27629099499374492049080296653, 1.12192424498000645566301779800, 4.08111424099919375876892903888, 5.35603503100533143843514964735, 6.53050885999447860292698284118, 7.49795253107976412906088376305, 8.983369314072910311456652972084, 10.35393877213564706642984944461, 11.22713938458541825628975454490, 12.55063765140411391514066176698, 13.17830731193204968147651679879

Graph of the ZZ-function along the critical line