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Properties

Label 2-13e2-169.127-c1-0-5
Degree 22
Conductor 169169
Sign 0.991−0.131i0.991 - 0.131i
Analytic cond. 1.349471.34947
Root an. cond. 1.161661.16166
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  + (−2.81 + 0.113i)2-s + (0.467 + 1.61i)3-s + (5.93 − 0.479i)4-s + (1.59 − 0.605i)5-s + (−1.50 − 4.49i)6-s + (1.15 − 2.71i)7-s + (−11.0 + 1.34i)8-s + (0.147 − 0.0930i)9-s + (−4.43 + 1.88i)10-s + (1.00 − 1.58i)11-s + (3.55 + 9.36i)12-s + (−1.18 − 3.40i)13-s + (−2.95 + 7.78i)14-s + (1.72 + 2.29i)15-s + (19.3 − 3.13i)16-s + (−0.139 − 0.0595i)17-s + ⋯
L(s)  = 1  + (−1.99 + 0.0803i)2-s + (0.270 + 0.932i)3-s + (2.96 − 0.239i)4-s + (0.714 − 0.270i)5-s + (−0.612 − 1.83i)6-s + (0.437 − 1.02i)7-s + (−3.91 + 0.475i)8-s + (0.0490 − 0.0310i)9-s + (−1.40 + 0.597i)10-s + (0.302 − 0.477i)11-s + (1.02 + 2.70i)12-s + (−0.329 − 0.944i)13-s + (−0.788 + 2.08i)14-s + (0.445 + 0.592i)15-s + (4.82 − 0.784i)16-s + (−0.0338 − 0.0144i)17-s + ⋯

Functional equation

Λ(s)=(169s/2 ΓC(s) L(s)=((0.991−0.131i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.131i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(169s/2 ΓC(s+1/2) L(s)=((0.991−0.131i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 - 0.131i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 169169    =    13213^{2}
Sign: 0.991−0.131i0.991 - 0.131i
Analytic conductor: 1.349471.34947
Root analytic conductor: 1.161661.16166
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ169(127,⋅)\chi_{169} (127, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 169, ( :1/2), 0.991−0.131i)(2,\ 169,\ (\ :1/2),\ 0.991 - 0.131i)

Particular Values

L(1)L(1) ≈\approx 0.650002+0.0429729i0.650002 + 0.0429729i
L(12)L(\frac12) ≈\approx 0.650002+0.0429729i0.650002 + 0.0429729i
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)
bad13 1+(1.18+3.40i)T 1 + (1.18 + 3.40i)T
good2 1+(2.81−0.113i)T+(1.99−0.160i)T2 1 + (2.81 - 0.113i)T + (1.99 - 0.160i)T^{2}
3 1+(−0.467−1.61i)T+(−2.53+1.60i)T2 1 + (-0.467 - 1.61i)T + (-2.53 + 1.60i)T^{2}
5 1+(−1.59+0.605i)T+(3.74−3.31i)T2 1 + (-1.59 + 0.605i)T + (3.74 - 3.31i)T^{2}
7 1+(−1.15+2.71i)T+(−4.84−5.04i)T2 1 + (-1.15 + 2.71i)T + (-4.84 - 5.04i)T^{2}
11 1+(−1.00+1.58i)T+(−4.71−9.93i)T2 1 + (-1.00 + 1.58i)T + (-4.71 - 9.93i)T^{2}
17 1+(0.139+0.0595i)T+(11.7+12.2i)T2 1 + (0.139 + 0.0595i)T + (11.7 + 12.2i)T^{2}
19 1+(−3.60−2.08i)T+(9.5+16.4i)T2 1 + (-3.60 - 2.08i)T + (9.5 + 16.4i)T^{2}
23 1+(0.549+0.951i)T+(−11.5+19.9i)T2 1 + (0.549 + 0.951i)T + (-11.5 + 19.9i)T^{2}
29 1+(−0.145−3.60i)T+(−28.9+2.33i)T2 1 + (-0.145 - 3.60i)T + (-28.9 + 2.33i)T^{2}
31 1+(3.40−3.84i)T+(−3.73−30.7i)T2 1 + (3.40 - 3.84i)T + (-3.73 - 30.7i)T^{2}
37 1+(3.51+0.717i)T+(34.0+14.5i)T2 1 + (3.51 + 0.717i)T + (34.0 + 14.5i)T^{2}
41 1+(4.62−1.34i)T+(34.6−21.9i)T2 1 + (4.62 - 1.34i)T + (34.6 - 21.9i)T^{2}
43 1+(−0.658−3.22i)T+(−39.5+16.8i)T2 1 + (-0.658 - 3.22i)T + (-39.5 + 16.8i)T^{2}
47 1+(−7.83−5.40i)T+(16.6+43.9i)T2 1 + (-7.83 - 5.40i)T + (16.6 + 43.9i)T^{2}
53 1+(0.943+7.76i)T+(−51.4+12.6i)T2 1 + (0.943 + 7.76i)T + (-51.4 + 12.6i)T^{2}
59 1+(−0.889+5.47i)T+(−55.9−18.6i)T2 1 + (-0.889 + 5.47i)T + (-55.9 - 18.6i)T^{2}
61 1+(8.26+6.21i)T+(16.9+58.5i)T2 1 + (8.26 + 6.21i)T + (16.9 + 58.5i)T^{2}
67 1+(1.16−14.4i)T+(−66.1−10.7i)T2 1 + (1.16 - 14.4i)T + (-66.1 - 10.7i)T^{2}
71 1+(3.77+3.62i)T+(2.85+70.9i)T2 1 + (3.77 + 3.62i)T + (2.85 + 70.9i)T^{2}
73 1+(0.450+0.857i)T+(−41.4+60.0i)T2 1 + (0.450 + 0.857i)T + (-41.4 + 60.0i)T^{2}
79 1+(0.951−1.37i)T+(−28.0−73.8i)T2 1 + (0.951 - 1.37i)T + (-28.0 - 73.8i)T^{2}
83 1+(−2.87−11.6i)T+(−73.4+38.5i)T2 1 + (-2.87 - 11.6i)T + (-73.4 + 38.5i)T^{2}
89 1+(−4.63+2.67i)T+(44.5−77.0i)T2 1 + (-4.63 + 2.67i)T + (44.5 - 77.0i)T^{2}
97 1+(6.11−4.99i)T+(19.4−95.0i)T2 1 + (6.11 - 4.99i)T + (19.4 - 95.0i)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

−12.44373009086945728267557515072, −11.10126590250307825760340306237, −10.39169621825813320957257950888, −9.777747466113841002943041045894, −8.981176390255912437388051659011, −7.939861154170715161821584247225, −6.95036325655286188234252953919, −5.50423351128108344195760150065, −3.31759687324944794412165861586, −1.31598885904049388558152219290, 1.74524081161765026228300052735, 2.42038113875547174657793257599, 5.90792126818545123972924995665, 6.95481225799952250236802754420, 7.67034193398040676638364435235, 8.826779852648619347797072115989, 9.477202908702137832203874224508, 10.46491752486020040012959290559, 11.80908103752512245586135486808, 12.12434204328674350757886720327

Graph of the ZZ-function along the critical line