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Properties

Label 2-136-17.15-c3-0-8
Degree 22
Conductor 136136
Sign −0.129+0.991i-0.129 + 0.991i
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  + (0.603 + 1.45i)3-s + (−10.3 + 4.29i)5-s + (−13.2 − 5.47i)7-s + (17.3 − 17.3i)9-s + (16.3 − 39.5i)11-s − 70.4i·13-s + (−12.5 − 12.5i)15-s + (−52.1 + 46.8i)17-s + (25.9 + 25.9i)19-s − 22.5i·21-s + (69.8 − 168. i)23-s + (0.842 − 0.842i)25-s + (75.0 + 31.0i)27-s + (−244. + 101. i)29-s + (−130. − 315. i)31-s + ⋯
L(s)  = 1  + (0.116 + 0.280i)3-s + (−0.928 + 0.384i)5-s + (−0.713 − 0.295i)7-s + (0.642 − 0.642i)9-s + (0.449 − 1.08i)11-s − 1.50i·13-s + (−0.215 − 0.215i)15-s + (−0.744 + 0.667i)17-s + (0.312 + 0.312i)19-s − 0.234i·21-s + (0.633 − 1.52i)23-s + (0.00674 − 0.00674i)25-s + (0.534 + 0.221i)27-s + (−1.56 + 0.648i)29-s + (−0.757 − 1.82i)31-s + ⋯

Functional equation

Λ(s)=(136s/2 ΓC(s) L(s)=((−0.129+0.991i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.129 + 0.991i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(136s/2 ΓC(s+3/2) L(s)=((−0.129+0.991i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.129 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 136136    =    23⋅172^{3} \cdot 17
Sign: −0.129+0.991i-0.129 + 0.991i
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), −0.129+0.991i)(2,\ 136,\ (\ :3/2),\ -0.129 + 0.991i)

Particular Values

L(2)L(2) ≈\approx 0.611161−0.696178i0.611161 - 0.696178i
L(12)L(\frac12) ≈\approx 0.611161−0.696178i0.611161 - 0.696178i
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+(52.1−46.8i)T 1 + (52.1 - 46.8i)T
good3 1+(−0.603−1.45i)T+(−19.0+19.0i)T2 1 + (-0.603 - 1.45i)T + (-19.0 + 19.0i)T^{2}
5 1+(10.3−4.29i)T+(88.3−88.3i)T2 1 + (10.3 - 4.29i)T + (88.3 - 88.3i)T^{2}
7 1+(13.2+5.47i)T+(242.+242.i)T2 1 + (13.2 + 5.47i)T + (242. + 242. i)T^{2}
11 1+(−16.3+39.5i)T+(−941.−941.i)T2 1 + (-16.3 + 39.5i)T + (-941. - 941. i)T^{2}
13 1+70.4iT−2.19e3T2 1 + 70.4iT - 2.19e3T^{2}
19 1+(−25.9−25.9i)T+6.85e3iT2 1 + (-25.9 - 25.9i)T + 6.85e3iT^{2}
23 1+(−69.8+168.i)T+(−8.60e3−8.60e3i)T2 1 + (-69.8 + 168. i)T + (-8.60e3 - 8.60e3i)T^{2}
29 1+(244.−101.i)T+(1.72e4−1.72e4i)T2 1 + (244. - 101. i)T + (1.72e4 - 1.72e4i)T^{2}
31 1+(130.+315.i)T+(−2.10e4+2.10e4i)T2 1 + (130. + 315. i)T + (-2.10e4 + 2.10e4i)T^{2}
37 1+(−25.0−60.5i)T+(−3.58e4+3.58e4i)T2 1 + (-25.0 - 60.5i)T + (-3.58e4 + 3.58e4i)T^{2}
41 1+(−13.9−5.79i)T+(4.87e4+4.87e4i)T2 1 + (-13.9 - 5.79i)T + (4.87e4 + 4.87e4i)T^{2}
43 1+(105.−105.i)T−7.95e4iT2 1 + (105. - 105. i)T - 7.95e4iT^{2}
47 1−4.93iT−1.03e5T2 1 - 4.93iT - 1.03e5T^{2}
53 1+(−465.−465.i)T+1.48e5iT2 1 + (-465. - 465. i)T + 1.48e5iT^{2}
59 1+(−74.6+74.6i)T−2.05e5iT2 1 + (-74.6 + 74.6i)T - 2.05e5iT^{2}
61 1+(543.+225.i)T+(1.60e5+1.60e5i)T2 1 + (543. + 225. i)T + (1.60e5 + 1.60e5i)T^{2}
67 1−952.T+3.00e5T2 1 - 952.T + 3.00e5T^{2}
71 1+(−127.−307.i)T+(−2.53e5+2.53e5i)T2 1 + (-127. - 307. i)T + (-2.53e5 + 2.53e5i)T^{2}
73 1+(622.−257.i)T+(2.75e5−2.75e5i)T2 1 + (622. - 257. i)T + (2.75e5 - 2.75e5i)T^{2}
79 1+(−201.+485.i)T+(−3.48e5−3.48e5i)T2 1 + (-201. + 485. i)T + (-3.48e5 - 3.48e5i)T^{2}
83 1+(−261.−261.i)T+5.71e5iT2 1 + (-261. - 261. i)T + 5.71e5iT^{2}
89 1+234.iT−7.04e5T2 1 + 234. iT - 7.04e5T^{2}
97 1+(−124.+51.6i)T+(6.45e5−6.45e5i)T2 1 + (-124. + 51.6i)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

−12.61991431361664052785518366584, −11.28020357553672929658723856456, −10.53237551396779386361749893041, −9.357797306709748617377686027350, −8.199278584045645242571573662134, −7.06471072207897694028523844211, −5.92080218602436813260003803948, −4.00950468734708596554545666179, −3.25544414888574037749824464095, −0.46192176494733000515730423585, 1.86873701681415399533054394417, 3.85172145580690973100466175822, 4.95128248220207634394375442422, 6.89418274325112143207845863758, 7.40962656591535551201473446972, 8.971566671026969373347404236271, 9.657047544148798332312554146058, 11.25808359361279536884741477833, 12.02525959115736845226104309139, 12.95087261366595356207402528010

Graph of the ZZ-function along the critical line