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Properties

Label 2-152-152.101-c1-0-8
Degree 22
Conductor 152152
Sign −0.710−0.703i-0.710 - 0.703i
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  + (0.717 + 1.21i)2-s + (−1.08 + 2.98i)3-s + (−0.970 + 1.74i)4-s + (3.40 − 0.600i)5-s + (−4.42 + 0.818i)6-s + (1.17 − 2.02i)7-s + (−2.82 + 0.0717i)8-s + (−5.45 − 4.57i)9-s + (3.17 + 3.72i)10-s + (1.02 − 0.592i)11-s + (−4.17 − 4.80i)12-s + (−0.845 − 2.32i)13-s + (3.31 − 0.0280i)14-s + (−1.91 + 10.8i)15-s + (−2.11 − 3.39i)16-s + (1.96 − 1.65i)17-s + ⋯
L(s)  = 1  + (0.507 + 0.861i)2-s + (−0.628 + 1.72i)3-s + (−0.485 + 0.874i)4-s + (1.52 − 0.268i)5-s + (−1.80 + 0.334i)6-s + (0.442 − 0.767i)7-s + (−0.999 + 0.0253i)8-s + (−1.81 − 1.52i)9-s + (1.00 + 1.17i)10-s + (0.309 − 0.178i)11-s + (−1.20 − 1.38i)12-s + (−0.234 − 0.644i)13-s + (0.885 − 0.00748i)14-s + (−0.493 + 2.79i)15-s + (−0.528 − 0.848i)16-s + (0.476 − 0.400i)17-s + ⋯

Functional equation

Λ(s)=(152s/2 ΓC(s) L(s)=((−0.710−0.703i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.710 - 0.703i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(152s/2 ΓC(s+1/2) L(s)=((−0.710−0.703i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.710 - 0.703i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 152152    =    23⋅192^{3} \cdot 19
Sign: −0.710−0.703i-0.710 - 0.703i
Analytic conductor: 1.213721.21372
Root analytic conductor: 1.101691.10169
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ152(101,⋅)\chi_{152} (101, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 152, ( :1/2), −0.710−0.703i)(2,\ 152,\ (\ :1/2),\ -0.710 - 0.703i)

Particular Values

L(1)L(1) ≈\approx 0.519761+1.26264i0.519761 + 1.26264i
L(12)L(\frac12) ≈\approx 0.519761+1.26264i0.519761 + 1.26264i
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+(−0.717−1.21i)T 1 + (-0.717 - 1.21i)T
19 1+(2.52−3.55i)T 1 + (2.52 - 3.55i)T
good3 1+(1.08−2.98i)T+(−2.29−1.92i)T2 1 + (1.08 - 2.98i)T + (-2.29 - 1.92i)T^{2}
5 1+(−3.40+0.600i)T+(4.69−1.71i)T2 1 + (-3.40 + 0.600i)T + (4.69 - 1.71i)T^{2}
7 1+(−1.17+2.02i)T+(−3.5−6.06i)T2 1 + (-1.17 + 2.02i)T + (-3.5 - 6.06i)T^{2}
11 1+(−1.02+0.592i)T+(5.5−9.52i)T2 1 + (-1.02 + 0.592i)T + (5.5 - 9.52i)T^{2}
13 1+(0.845+2.32i)T+(−9.95+8.35i)T2 1 + (0.845 + 2.32i)T + (-9.95 + 8.35i)T^{2}
17 1+(−1.96+1.65i)T+(2.95−16.7i)T2 1 + (-1.96 + 1.65i)T + (2.95 - 16.7i)T^{2}
23 1+(−0.0208+0.118i)T+(−21.6−7.86i)T2 1 + (-0.0208 + 0.118i)T + (-21.6 - 7.86i)T^{2}
29 1+(3.07−3.65i)T+(−5.03−28.5i)T2 1 + (3.07 - 3.65i)T + (-5.03 - 28.5i)T^{2}
31 1+(−0.669+1.15i)T+(−15.5−26.8i)T2 1 + (-0.669 + 1.15i)T + (-15.5 - 26.8i)T^{2}
37 1−3.90iT−37T2 1 - 3.90iT - 37T^{2}
41 1+(1.30+0.474i)T+(31.4+26.3i)T2 1 + (1.30 + 0.474i)T + (31.4 + 26.3i)T^{2}
43 1+(9.95−1.75i)T+(40.4−14.7i)T2 1 + (9.95 - 1.75i)T + (40.4 - 14.7i)T^{2}
47 1+(2.76+2.31i)T+(8.16+46.2i)T2 1 + (2.76 + 2.31i)T + (8.16 + 46.2i)T^{2}
53 1+(5.70+1.00i)T+(49.8+18.1i)T2 1 + (5.70 + 1.00i)T + (49.8 + 18.1i)T^{2}
59 1+(5.32+6.34i)T+(−10.2+58.1i)T2 1 + (5.32 + 6.34i)T + (-10.2 + 58.1i)T^{2}
61 1+(−11.6−2.05i)T+(57.3+20.8i)T2 1 + (-11.6 - 2.05i)T + (57.3 + 20.8i)T^{2}
67 1+(−7.80+9.29i)T+(−11.6−65.9i)T2 1 + (-7.80 + 9.29i)T + (-11.6 - 65.9i)T^{2}
71 1+(−0.0402−0.228i)T+(−66.7+24.2i)T2 1 + (-0.0402 - 0.228i)T + (-66.7 + 24.2i)T^{2}
73 1+(−2.45−0.891i)T+(55.9+46.9i)T2 1 + (-2.45 - 0.891i)T + (55.9 + 46.9i)T^{2}
79 1+(−15.4−5.61i)T+(60.5+50.7i)T2 1 + (-15.4 - 5.61i)T + (60.5 + 50.7i)T^{2}
83 1+(6.76+3.90i)T+(41.5+71.8i)T2 1 + (6.76 + 3.90i)T + (41.5 + 71.8i)T^{2}
89 1+(6.23−2.27i)T+(68.1−57.2i)T2 1 + (6.23 - 2.27i)T + (68.1 - 57.2i)T^{2}
97 1+(12.0−10.1i)T+(16.8−95.5i)T2 1 + (12.0 - 10.1i)T + (16.8 - 95.5i)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

−13.74254324928631614093220874876, −12.52001256002444379023047089194, −11.17664704792845860328546280480, −10.08854304653782974276332380269, −9.522636855756292317906574280709, −8.313731819752573918957579622220, −6.46979321117945320623665142007, −5.48059217358989877544623321511, −4.83229349379335417041797759175, −3.51908843454143737163675252232, 1.67923192173290987390214577763, 2.39398764082280098202128572651, 5.18031254260211879136276956720, 6.01543584827256269666832250966, 6.80499615023528836096252622358, 8.570734395818371527202153744211, 9.731307329610796789152788254309, 11.01605530314907592443102563198, 11.80644535609681155901980123675, 12.65536618591934259406933948586

Graph of the ZZ-function along the critical line