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Properties

Label 2-136-17.15-c3-0-2
Degree 22
Conductor 136136
Sign −0.913−0.407i-0.913 - 0.407i
Analytic cond. 8.024258.02425
Root an. cond. 2.832712.83271
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (3.26 + 7.87i)3-s + (−8.68 + 3.59i)5-s + (3.82 + 1.58i)7-s + (−32.3 + 32.3i)9-s + (−0.766 + 1.85i)11-s + 58.7i·13-s + (−56.7 − 56.7i)15-s + (−29.9 − 63.3i)17-s + (−59.0 − 59.0i)19-s + 35.2i·21-s + (−15.0 + 36.2i)23-s + (−25.8 + 25.8i)25-s + (−147. − 61.1i)27-s + (145. − 60.4i)29-s + (46.0 + 111. i)31-s + ⋯
L(s)  = 1  + (0.628 + 1.51i)3-s + (−0.777 + 0.321i)5-s + (0.206 + 0.0855i)7-s + (−1.19 + 1.19i)9-s + (−0.0210 + 0.0507i)11-s + 1.25i·13-s + (−0.976 − 0.976i)15-s + (−0.427 − 0.904i)17-s + (−0.712 − 0.712i)19-s + 0.366i·21-s + (−0.136 + 0.329i)23-s + (−0.206 + 0.206i)25-s + (−1.05 − 0.435i)27-s + (0.934 − 0.386i)29-s + (0.266 + 0.643i)31-s + ⋯

Functional equation

Λ(s)=(136s/2 ΓC(s) L(s)=((−0.913−0.407i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.913 - 0.407i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(136s/2 ΓC(s+3/2) L(s)=((−0.913−0.407i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.913 - 0.407i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 136136    =    23⋅172^{3} \cdot 17
Sign: −0.913−0.407i-0.913 - 0.407i
Analytic conductor: 8.024258.02425
Root analytic conductor: 2.832712.83271
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ136(49,⋅)\chi_{136} (49, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 136, ( :3/2), −0.913−0.407i)(2,\ 136,\ (\ :3/2),\ -0.913 - 0.407i)

Particular Values

L(2)L(2) ≈\approx 0.295511+1.38735i0.295511 + 1.38735i
L(12)L(\frac12) ≈\approx 0.295511+1.38735i0.295511 + 1.38735i
L(52)L(\frac{5}{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
17 1+(29.9+63.3i)T 1 + (29.9 + 63.3i)T
good3 1+(−3.26−7.87i)T+(−19.0+19.0i)T2 1 + (-3.26 - 7.87i)T + (-19.0 + 19.0i)T^{2}
5 1+(8.68−3.59i)T+(88.3−88.3i)T2 1 + (8.68 - 3.59i)T + (88.3 - 88.3i)T^{2}
7 1+(−3.82−1.58i)T+(242.+242.i)T2 1 + (-3.82 - 1.58i)T + (242. + 242. i)T^{2}
11 1+(0.766−1.85i)T+(−941.−941.i)T2 1 + (0.766 - 1.85i)T + (-941. - 941. i)T^{2}
13 1−58.7iT−2.19e3T2 1 - 58.7iT - 2.19e3T^{2}
19 1+(59.0+59.0i)T+6.85e3iT2 1 + (59.0 + 59.0i)T + 6.85e3iT^{2}
23 1+(15.0−36.2i)T+(−8.60e3−8.60e3i)T2 1 + (15.0 - 36.2i)T + (-8.60e3 - 8.60e3i)T^{2}
29 1+(−145.+60.4i)T+(1.72e4−1.72e4i)T2 1 + (-145. + 60.4i)T + (1.72e4 - 1.72e4i)T^{2}
31 1+(−46.0−111.i)T+(−2.10e4+2.10e4i)T2 1 + (-46.0 - 111. i)T + (-2.10e4 + 2.10e4i)T^{2}
37 1+(−129.−312.i)T+(−3.58e4+3.58e4i)T2 1 + (-129. - 312. i)T + (-3.58e4 + 3.58e4i)T^{2}
41 1+(−251.−104.i)T+(4.87e4+4.87e4i)T2 1 + (-251. - 104. i)T + (4.87e4 + 4.87e4i)T^{2}
43 1+(169.−169.i)T−7.95e4iT2 1 + (169. - 169. i)T - 7.95e4iT^{2}
47 1−150.iT−1.03e5T2 1 - 150. iT - 1.03e5T^{2}
53 1+(−513.−513.i)T+1.48e5iT2 1 + (-513. - 513. i)T + 1.48e5iT^{2}
59 1+(−448.+448.i)T−2.05e5iT2 1 + (-448. + 448. i)T - 2.05e5iT^{2}
61 1+(−570.−236.i)T+(1.60e5+1.60e5i)T2 1 + (-570. - 236. i)T + (1.60e5 + 1.60e5i)T^{2}
67 1−712.T+3.00e5T2 1 - 712.T + 3.00e5T^{2}
71 1+(317.+767.i)T+(−2.53e5+2.53e5i)T2 1 + (317. + 767. i)T + (-2.53e5 + 2.53e5i)T^{2}
73 1+(471.−195.i)T+(2.75e5−2.75e5i)T2 1 + (471. - 195. i)T + (2.75e5 - 2.75e5i)T^{2}
79 1+(−114.+276.i)T+(−3.48e5−3.48e5i)T2 1 + (-114. + 276. i)T + (-3.48e5 - 3.48e5i)T^{2}
83 1+(434.+434.i)T+5.71e5iT2 1 + (434. + 434. i)T + 5.71e5iT^{2}
89 1−1.26e3iT−7.04e5T2 1 - 1.26e3iT - 7.04e5T^{2}
97 1+(949.−393.i)T+(6.45e5−6.45e5i)T2 1 + (949. - 393. i)T + (6.45e5 - 6.45e5i)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

−13.49422911048350686866362004181, −11.73079596486708792180761228411, −11.15699678286673566075917315710, −9.973419353813731543024883448846, −9.113571817396107189038443678301, −8.185542893088226274401437822714, −6.76924898903674223265353664726, −4.82766308002810200890720022481, −4.08627606781493887142545045052, −2.74772286507374031014954201841, 0.66275591091099976055946651244, 2.30345772826378292028453398866, 3.92869883159259643257289524300, 5.86832550367782564284911395474, 7.10379274189565165390859688796, 8.186283061269735516287268637117, 8.448796495828055524818462937597, 10.32029956276619069161231342688, 11.60590841908722389522372386687, 12.66172926560874339254450871621

Graph of the ZZ-function along the critical line