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Properties

Label 2-117-117.22-c1-0-0
Degree 22
Conductor 117117
Sign −0.952−0.304i-0.952 - 0.304i
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  − 2.28·2-s + (−0.409 + 1.68i)3-s + 3.21·4-s + (−0.461 + 0.799i)5-s + (0.935 − 3.84i)6-s + (−0.454 + 0.786i)7-s − 2.76·8-s + (−2.66 − 1.37i)9-s + (1.05 − 1.82i)10-s − 4.78·11-s + (−1.31 + 5.40i)12-s + (−2.54 + 2.54i)13-s + (1.03 − 1.79i)14-s + (−1.15 − 1.10i)15-s − 0.112·16-s + (1.98 + 3.44i)17-s + ⋯
L(s)  = 1  − 1.61·2-s + (−0.236 + 0.971i)3-s + 1.60·4-s + (−0.206 + 0.357i)5-s + (0.381 − 1.56i)6-s + (−0.171 + 0.297i)7-s − 0.977·8-s + (−0.888 − 0.459i)9-s + (0.333 − 0.576i)10-s − 1.44·11-s + (−0.379 + 1.55i)12-s + (−0.707 + 0.707i)13-s + (0.277 − 0.479i)14-s + (−0.298 − 0.285i)15-s − 0.0280·16-s + (0.481 + 0.834i)17-s + ⋯

Functional equation

Λ(s)=(117s/2 ΓC(s) L(s)=((−0.952−0.304i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.952 - 0.304i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(117s/2 ΓC(s+1/2) L(s)=((−0.952−0.304i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.952 - 0.304i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 117117    =    32⋅133^{2} \cdot 13
Sign: −0.952−0.304i-0.952 - 0.304i
Analytic conductor: 0.9342490.934249
Root analytic conductor: 0.9665650.966565
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ117(22,⋅)\chi_{117} (22, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 117, ( :1/2), −0.952−0.304i)(2,\ 117,\ (\ :1/2),\ -0.952 - 0.304i)

Particular Values

L(1)L(1) ≈\approx 0.0388349+0.248767i0.0388349 + 0.248767i
L(12)L(\frac12) ≈\approx 0.0388349+0.248767i0.0388349 + 0.248767i
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+(0.409−1.68i)T 1 + (0.409 - 1.68i)T
13 1+(2.54−2.54i)T 1 + (2.54 - 2.54i)T
good2 1+2.28T+2T2 1 + 2.28T + 2T^{2}
5 1+(0.461−0.799i)T+(−2.5−4.33i)T2 1 + (0.461 - 0.799i)T + (-2.5 - 4.33i)T^{2}
7 1+(0.454−0.786i)T+(−3.5−6.06i)T2 1 + (0.454 - 0.786i)T + (-3.5 - 6.06i)T^{2}
11 1+4.78T+11T2 1 + 4.78T + 11T^{2}
17 1+(−1.98−3.44i)T+(−8.5+14.7i)T2 1 + (-1.98 - 3.44i)T + (-8.5 + 14.7i)T^{2}
19 1+(2.47+4.29i)T+(−9.5+16.4i)T2 1 + (2.47 + 4.29i)T + (-9.5 + 16.4i)T^{2}
23 1+(0.489+0.848i)T+(−11.5+19.9i)T2 1 + (0.489 + 0.848i)T + (-11.5 + 19.9i)T^{2}
29 1−6.06T+29T2 1 - 6.06T + 29T^{2}
31 1+(2.18−3.79i)T+(−15.5−26.8i)T2 1 + (2.18 - 3.79i)T + (-15.5 - 26.8i)T^{2}
37 1+(0.728−1.26i)T+(−18.5−32.0i)T2 1 + (0.728 - 1.26i)T + (-18.5 - 32.0i)T^{2}
41 1+(−0.648−1.12i)T+(−20.5+35.5i)T2 1 + (-0.648 - 1.12i)T + (-20.5 + 35.5i)T^{2}
43 1+(3.69−6.40i)T+(−21.5−37.2i)T2 1 + (3.69 - 6.40i)T + (-21.5 - 37.2i)T^{2}
47 1+(−4.74−8.21i)T+(−23.5+40.7i)T2 1 + (-4.74 - 8.21i)T + (-23.5 + 40.7i)T^{2}
53 1+10.0T+53T2 1 + 10.0T + 53T^{2}
59 1−2.31T+59T2 1 - 2.31T + 59T^{2}
61 1+(−6.26+10.8i)T+(−30.5−52.8i)T2 1 + (-6.26 + 10.8i)T + (-30.5 - 52.8i)T^{2}
67 1+(−0.780−1.35i)T+(−33.5+58.0i)T2 1 + (-0.780 - 1.35i)T + (-33.5 + 58.0i)T^{2}
71 1+(−4.58−7.93i)T+(−35.5+61.4i)T2 1 + (-4.58 - 7.93i)T + (-35.5 + 61.4i)T^{2}
73 1−14.0T+73T2 1 - 14.0T + 73T^{2}
79 1+(4.86+8.42i)T+(−39.5+68.4i)T2 1 + (4.86 + 8.42i)T + (-39.5 + 68.4i)T^{2}
83 1+(8.11+14.0i)T+(−41.5+71.8i)T2 1 + (8.11 + 14.0i)T + (-41.5 + 71.8i)T^{2}
89 1+(2.29−3.97i)T+(−44.5−77.0i)T2 1 + (2.29 - 3.97i)T + (-44.5 - 77.0i)T^{2}
97 1+(4.09−7.09i)T+(−48.5−84.0i)T2 1 + (4.09 - 7.09i)T + (-48.5 - 84.0i)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

−14.39748144942388136245595544455, −12.64425768861313622023991358105, −11.28346415647624988836367331880, −10.61824974145615845154231691386, −9.813644027633977810323562252535, −8.858345806740065783742774480001, −7.85396496283906490546063474928, −6.53243042042171104954018010297, −4.86211635149944120754238452662, −2.75220844010313122783680381913, 0.44335391046638664084237195874, 2.44589095841391484067598140695, 5.36081485191582954317488477256, 6.95449861688529875077582421448, 7.84883190883476050827203271345, 8.435736871538288378407207748004, 9.982107901651616056358787906623, 10.65459936798260525731162241520, 11.94776304128397456229467402479, 12.79052994657381967614980134673

Graph of the ZZ-function along the critical line