↑

Properties

Label 2-136-17.9-c3-0-8
Degree 22
Conductor 136136
Sign 0.890−0.454i0.890 - 0.454i
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (8.27 + 3.42i)3-s + (1.41 − 3.41i)5-s + (−0.563 − 1.36i)7-s + (37.6 + 37.6i)9-s + (54.6 − 22.6i)11-s + 12.0i·13-s + (23.3 − 23.3i)15-s + (−63.8 + 28.9i)17-s + (−29.8 + 29.8i)19-s − 13.1i·21-s + (−91.5 + 37.9i)23-s + (78.7 + 78.7i)25-s + (90.0 + 217. i)27-s + (97.5 − 235. i)29-s + (−155. − 64.5i)31-s + ⋯
L(s)  = 1  + (1.59 + 0.659i)3-s + (0.126 − 0.305i)5-s + (−0.0304 − 0.0734i)7-s + (1.39 + 1.39i)9-s + (1.49 − 0.620i)11-s + 0.257i·13-s + (0.402 − 0.402i)15-s + (−0.910 + 0.413i)17-s + (−0.360 + 0.360i)19-s − 0.137i·21-s + (−0.829 + 0.343i)23-s + (0.629 + 0.629i)25-s + (0.642 + 1.55i)27-s + (0.624 − 1.50i)29-s + (−0.903 − 0.374i)31-s + ⋯

Functional equation

Λ(s)=(136s/2 ΓC(s) L(s)=((0.890−0.454i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.890 - 0.454i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(136s/2 ΓC(s+3/2) L(s)=((0.890−0.454i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.890 - 0.454i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 136136    =    23⋅172^{3} \cdot 17
Sign: 0.890−0.454i0.890 - 0.454i
Analytic conductor: 8.024258.02425
Root analytic conductor: 2.832712.83271
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ136(9,⋅)\chi_{136} (9, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 136, ( :3/2), 0.890−0.454i)(2,\ 136,\ (\ :3/2),\ 0.890 - 0.454i)

Particular Values

L(2)L(2) ≈\approx 2.76283+0.664846i2.76283 + 0.664846i
L(12)L(\frac12) ≈\approx 2.76283+0.664846i2.76283 + 0.664846i
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+(63.8−28.9i)T 1 + (63.8 - 28.9i)T
good3 1+(−8.27−3.42i)T+(19.0+19.0i)T2 1 + (-8.27 - 3.42i)T + (19.0 + 19.0i)T^{2}
5 1+(−1.41+3.41i)T+(−88.3−88.3i)T2 1 + (-1.41 + 3.41i)T + (-88.3 - 88.3i)T^{2}
7 1+(0.563+1.36i)T+(−242.+242.i)T2 1 + (0.563 + 1.36i)T + (-242. + 242. i)T^{2}
11 1+(−54.6+22.6i)T+(941.−941.i)T2 1 + (-54.6 + 22.6i)T + (941. - 941. i)T^{2}
13 1−12.0iT−2.19e3T2 1 - 12.0iT - 2.19e3T^{2}
19 1+(29.8−29.8i)T−6.85e3iT2 1 + (29.8 - 29.8i)T - 6.85e3iT^{2}
23 1+(91.5−37.9i)T+(8.60e3−8.60e3i)T2 1 + (91.5 - 37.9i)T + (8.60e3 - 8.60e3i)T^{2}
29 1+(−97.5+235.i)T+(−1.72e4−1.72e4i)T2 1 + (-97.5 + 235. i)T + (-1.72e4 - 1.72e4i)T^{2}
31 1+(155.+64.5i)T+(2.10e4+2.10e4i)T2 1 + (155. + 64.5i)T + (2.10e4 + 2.10e4i)T^{2}
37 1+(70.9+29.3i)T+(3.58e4+3.58e4i)T2 1 + (70.9 + 29.3i)T + (3.58e4 + 3.58e4i)T^{2}
41 1+(−18.0−43.5i)T+(−4.87e4+4.87e4i)T2 1 + (-18.0 - 43.5i)T + (-4.87e4 + 4.87e4i)T^{2}
43 1+(160.+160.i)T+7.95e4iT2 1 + (160. + 160. i)T + 7.95e4iT^{2}
47 1−279.iT−1.03e5T2 1 - 279. iT - 1.03e5T^{2}
53 1+(149.−149.i)T−1.48e5iT2 1 + (149. - 149. i)T - 1.48e5iT^{2}
59 1+(74.5+74.5i)T+2.05e5iT2 1 + (74.5 + 74.5i)T + 2.05e5iT^{2}
61 1+(316.+763.i)T+(−1.60e5+1.60e5i)T2 1 + (316. + 763. i)T + (-1.60e5 + 1.60e5i)T^{2}
67 1+638.T+3.00e5T2 1 + 638.T + 3.00e5T^{2}
71 1+(998.+413.i)T+(2.53e5+2.53e5i)T2 1 + (998. + 413. i)T + (2.53e5 + 2.53e5i)T^{2}
73 1+(54.5−131.i)T+(−2.75e5−2.75e5i)T2 1 + (54.5 - 131. i)T + (-2.75e5 - 2.75e5i)T^{2}
79 1+(−201.+83.5i)T+(3.48e5−3.48e5i)T2 1 + (-201. + 83.5i)T + (3.48e5 - 3.48e5i)T^{2}
83 1+(−883.+883.i)T−5.71e5iT2 1 + (-883. + 883. i)T - 5.71e5iT^{2}
89 1−1.22e3iT−7.04e5T2 1 - 1.22e3iT - 7.04e5T^{2}
97 1+(−66.0+159.i)T+(−6.45e5−6.45e5i)T2 1 + (-66.0 + 159. i)T + (-6.45e5 - 6.45e5i)T^{2}
show more
show less
   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

−13.20502261165679538181492056544, −11.80748094629664285651979188684, −10.51547690282771894755642147134, −9.316325633936317560278652970944, −8.882724852891469874639099390320, −7.82589615704573877125624741371, −6.31253122800649645168967111386, −4.38140713455173487092718738816, −3.53668594330003550299124755597, −1.89107731094651368842411071319, 1.66196697341950403136447114777, 2.94077269203782435481902931782, 4.29063459103099662050278356005, 6.57406523641069487627515576302, 7.24000635626940995761270988501, 8.651114959647365775957796157739, 9.132946346234672787727792412096, 10.40528351122980092587849984924, 11.96555850760396292914586567445, 12.82094131406798306824906827034

Graph of the ZZ-function along the critical line