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Properties

Label 2-116-29.24-c1-0-0
Degree 22
Conductor 116116
Sign 0.332−0.943i0.332 - 0.943i
Analytic cond. 0.9262640.926264
Root an. cond. 0.9624260.962426
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  + (0.588 + 2.58i)3-s + (−0.379 − 0.475i)5-s + (0.143 + 0.630i)7-s + (−3.60 + 1.73i)9-s + (2.86 + 1.37i)11-s + (−4.97 − 2.39i)13-s + (1.00 − 1.25i)15-s + 6.66·17-s + (1.14 − 5.02i)19-s + (−1.54 + 0.742i)21-s + (1.01 − 1.27i)23-s + (1.03 − 4.51i)25-s + (−1.65 − 2.08i)27-s + (−5.38 + 0.122i)29-s + (−1.23 − 1.54i)31-s + ⋯
L(s)  = 1  + (0.340 + 1.48i)3-s + (−0.169 − 0.212i)5-s + (0.0544 + 0.238i)7-s + (−1.20 + 0.579i)9-s + (0.863 + 0.415i)11-s + (−1.37 − 0.664i)13-s + (0.259 − 0.325i)15-s + 1.61·17-s + (0.262 − 1.15i)19-s + (−0.336 + 0.162i)21-s + (0.211 − 0.265i)23-s + (0.206 − 0.902i)25-s + (−0.319 − 0.400i)27-s + (−0.999 + 0.0226i)29-s + (−0.221 − 0.277i)31-s + ⋯

Functional equation

Λ(s)=(116s/2 ΓC(s) L(s)=((0.332−0.943i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 116 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.332 - 0.943i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(116s/2 ΓC(s+1/2) L(s)=((0.332−0.943i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 116 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.332 - 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 116116    =    22⋅292^{2} \cdot 29
Sign: 0.332−0.943i0.332 - 0.943i
Analytic conductor: 0.9262640.926264
Root analytic conductor: 0.9624260.962426
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ116(53,⋅)\chi_{116} (53, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 116, ( :1/2), 0.332−0.943i)(2,\ 116,\ (\ :1/2),\ 0.332 - 0.943i)

Particular Values

L(1)L(1) ≈\approx 0.920018+0.651283i0.920018 + 0.651283i
L(12)L(\frac12) ≈\approx 0.920018+0.651283i0.920018 + 0.651283i
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)
bad2 1 1
29 1+(5.38−0.122i)T 1 + (5.38 - 0.122i)T
good3 1+(−0.588−2.58i)T+(−2.70+1.30i)T2 1 + (-0.588 - 2.58i)T + (-2.70 + 1.30i)T^{2}
5 1+(0.379+0.475i)T+(−1.11+4.87i)T2 1 + (0.379 + 0.475i)T + (-1.11 + 4.87i)T^{2}
7 1+(−0.143−0.630i)T+(−6.30+3.03i)T2 1 + (-0.143 - 0.630i)T + (-6.30 + 3.03i)T^{2}
11 1+(−2.86−1.37i)T+(6.85+8.60i)T2 1 + (-2.86 - 1.37i)T + (6.85 + 8.60i)T^{2}
13 1+(4.97+2.39i)T+(8.10+10.1i)T2 1 + (4.97 + 2.39i)T + (8.10 + 10.1i)T^{2}
17 1−6.66T+17T2 1 - 6.66T + 17T^{2}
19 1+(−1.14+5.02i)T+(−17.1−8.24i)T2 1 + (-1.14 + 5.02i)T + (-17.1 - 8.24i)T^{2}
23 1+(−1.01+1.27i)T+(−5.11−22.4i)T2 1 + (-1.01 + 1.27i)T + (-5.11 - 22.4i)T^{2}
31 1+(1.23+1.54i)T+(−6.89+30.2i)T2 1 + (1.23 + 1.54i)T + (-6.89 + 30.2i)T^{2}
37 1+(2.60−1.25i)T+(23.0−28.9i)T2 1 + (2.60 - 1.25i)T + (23.0 - 28.9i)T^{2}
41 1+9.83T+41T2 1 + 9.83T + 41T^{2}
43 1+(4.81−6.04i)T+(−9.56−41.9i)T2 1 + (4.81 - 6.04i)T + (-9.56 - 41.9i)T^{2}
47 1+(−7.49−3.61i)T+(29.3+36.7i)T2 1 + (-7.49 - 3.61i)T + (29.3 + 36.7i)T^{2}
53 1+(−4.82−6.05i)T+(−11.7+51.6i)T2 1 + (-4.82 - 6.05i)T + (-11.7 + 51.6i)T^{2}
59 1+1.12T+59T2 1 + 1.12T + 59T^{2}
61 1+(2.02+8.85i)T+(−54.9+26.4i)T2 1 + (2.02 + 8.85i)T + (-54.9 + 26.4i)T^{2}
67 1+(11.2−5.42i)T+(41.7−52.3i)T2 1 + (11.2 - 5.42i)T + (41.7 - 52.3i)T^{2}
71 1+(−0.344−0.165i)T+(44.2+55.5i)T2 1 + (-0.344 - 0.165i)T + (44.2 + 55.5i)T^{2}
73 1+(−0.200+0.250i)T+(−16.2−71.1i)T2 1 + (-0.200 + 0.250i)T + (-16.2 - 71.1i)T^{2}
79 1+(15.5−7.46i)T+(49.2−61.7i)T2 1 + (15.5 - 7.46i)T + (49.2 - 61.7i)T^{2}
83 1+(1.91−8.39i)T+(−74.7−36.0i)T2 1 + (1.91 - 8.39i)T + (-74.7 - 36.0i)T^{2}
89 1+(5.99+7.51i)T+(−19.8+86.7i)T2 1 + (5.99 + 7.51i)T + (-19.8 + 86.7i)T^{2}
97 1+(−1.95+8.56i)T+(−87.3−42.0i)T2 1 + (-1.95 + 8.56i)T + (-87.3 - 42.0i)T^{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

−14.15906019297343224947635640194, −12.55607819908128685685049287171, −11.64958639859171303168326599123, −10.28434268190448235075191660809, −9.659829616828704386905523312871, −8.701929995565410148744078901157, −7.32602151187432309692594152223, −5.39850615359110643922123513494, −4.43772895864940991528890152682, −3.04659356474381114692215357095, 1.62649923129061217627517206619, 3.46354064703133893143755187244, 5.62512948390626350713036766625, 7.08050560975159264703435246568, 7.54032610327219811266182545257, 8.863210735976551177042467678812, 10.15191452186472313903367142981, 11.87522369886409150422479371865, 12.12165386986212495239156780142, 13.41195986842536371728567642850

Graph of the ZZ-function along the critical line