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Properties

Label 2-152-152.107-c1-0-4
Degree 22
Conductor 152152
Sign −0.124−0.992i-0.124 - 0.992i
Analytic cond. 1.213721.21372
Root an. cond. 1.101691.10169
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  + (1.41 − 0.0756i)2-s + (−2.65 + 1.53i)3-s + (1.98 − 0.213i)4-s + (−2.25 + 1.30i)5-s + (−3.63 + 2.36i)6-s + 4.30i·7-s + (2.79 − 0.452i)8-s + (3.20 − 5.54i)9-s + (−3.08 + 2.01i)10-s − 0.349·11-s + (−4.95 + 3.61i)12-s + (0.839 − 1.45i)13-s + (0.325 + 6.07i)14-s + (3.99 − 6.92i)15-s + (3.90 − 0.850i)16-s + (0.357 + 0.618i)17-s + ⋯
L(s)  = 1  + (0.998 − 0.0535i)2-s + (−1.53 + 0.885i)3-s + (0.994 − 0.106i)4-s + (−1.00 + 0.582i)5-s + (−1.48 + 0.965i)6-s + 1.62i·7-s + (0.987 − 0.159i)8-s + (1.06 − 1.84i)9-s + (−0.976 + 0.635i)10-s − 0.105·11-s + (−1.42 + 1.04i)12-s + (0.232 − 0.403i)13-s + (0.0870 + 1.62i)14-s + (1.03 − 1.78i)15-s + (0.977 − 0.212i)16-s + (0.0865 + 0.149i)17-s + ⋯

Functional equation

Λ(s)=(152s/2 ΓC(s) L(s)=((−0.124−0.992i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.124 - 0.992i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(152s/2 ΓC(s+1/2) L(s)=((−0.124−0.992i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.124 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 152152    =    23⋅192^{3} \cdot 19
Sign: −0.124−0.992i-0.124 - 0.992i
Analytic conductor: 1.213721.21372
Root analytic conductor: 1.101691.10169
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ152(107,⋅)\chi_{152} (107, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 152, ( :1/2), −0.124−0.992i)(2,\ 152,\ (\ :1/2),\ -0.124 - 0.992i)

Particular Values

L(1)L(1) ≈\approx 0.728185+0.824858i0.728185 + 0.824858i
L(12)L(\frac12) ≈\approx 0.728185+0.824858i0.728185 + 0.824858i
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+(−1.41+0.0756i)T 1 + (-1.41 + 0.0756i)T
19 1+(−4.35+0.268i)T 1 + (-4.35 + 0.268i)T
good3 1+(2.65−1.53i)T+(1.5−2.59i)T2 1 + (2.65 - 1.53i)T + (1.5 - 2.59i)T^{2}
5 1+(2.25−1.30i)T+(2.5−4.33i)T2 1 + (2.25 - 1.30i)T + (2.5 - 4.33i)T^{2}
7 1−4.30iT−7T2 1 - 4.30iT - 7T^{2}
11 1+0.349T+11T2 1 + 0.349T + 11T^{2}
13 1+(−0.839+1.45i)T+(−6.5−11.2i)T2 1 + (-0.839 + 1.45i)T + (-6.5 - 11.2i)T^{2}
17 1+(−0.357−0.618i)T+(−8.5+14.7i)T2 1 + (-0.357 - 0.618i)T + (-8.5 + 14.7i)T^{2}
23 1+(1.38+0.797i)T+(11.5+19.9i)T2 1 + (1.38 + 0.797i)T + (11.5 + 19.9i)T^{2}
29 1+(−0.463+0.803i)T+(−14.5−25.1i)T2 1 + (-0.463 + 0.803i)T + (-14.5 - 25.1i)T^{2}
31 1+2.80T+31T2 1 + 2.80T + 31T^{2}
37 1−10.2T+37T2 1 - 10.2T + 37T^{2}
41 1+(−4.87+2.81i)T+(20.5−35.5i)T2 1 + (-4.87 + 2.81i)T + (20.5 - 35.5i)T^{2}
43 1+(−4.32−7.48i)T+(−21.5+37.2i)T2 1 + (-4.32 - 7.48i)T + (-21.5 + 37.2i)T^{2}
47 1+(6.26+3.61i)T+(23.5+40.7i)T2 1 + (6.26 + 3.61i)T + (23.5 + 40.7i)T^{2}
53 1+(4.73−8.20i)T+(−26.5−45.8i)T2 1 + (4.73 - 8.20i)T + (-26.5 - 45.8i)T^{2}
59 1+(−6.62+3.82i)T+(29.5−51.0i)T2 1 + (-6.62 + 3.82i)T + (29.5 - 51.0i)T^{2}
61 1+(3.46+2.00i)T+(30.5+52.8i)T2 1 + (3.46 + 2.00i)T + (30.5 + 52.8i)T^{2}
67 1+(10.6+6.12i)T+(33.5+58.0i)T2 1 + (10.6 + 6.12i)T + (33.5 + 58.0i)T^{2}
71 1+(5.09+8.82i)T+(−35.5+61.4i)T2 1 + (5.09 + 8.82i)T + (-35.5 + 61.4i)T^{2}
73 1+(−2.54−4.40i)T+(−36.5+63.2i)T2 1 + (-2.54 - 4.40i)T + (-36.5 + 63.2i)T^{2}
79 1+(1.31+2.27i)T+(−39.5+68.4i)T2 1 + (1.31 + 2.27i)T + (-39.5 + 68.4i)T^{2}
83 1+4.69T+83T2 1 + 4.69T + 83T^{2}
89 1+(−7.37−4.25i)T+(44.5+77.0i)T2 1 + (-7.37 - 4.25i)T + (44.5 + 77.0i)T^{2}
97 1+(−6.30+3.63i)T+(48.5−84.0i)T2 1 + (-6.30 + 3.63i)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

−12.80542095474111825965166752800, −11.99641092731616482602312294723, −11.46242512035260745658063168735, −10.79925316258895977062719451524, −9.544399062332076676955606714897, −7.70352351658690313531356906566, −6.23225247455229623948171538106, −5.58078426714627991885852290604, −4.48518694295945119368423262262, −3.15395029256111904059506390777, 1.04175276903995945481927236818, 3.98166891084998513451631797396, 4.88996538710843593277091480487, 6.16537333575942203225387311815, 7.30305998676960456153843613085, 7.71160625671602733460123175240, 10.27772712848495111290188076587, 11.32367061408005558402984489455, 11.68058095069953966327887904194, 12.74852636901948811135346104746

Graph of the ZZ-function along the critical line