Properties

Label 2-250-25.6-c3-0-1
Degree 22
Conductor 250250
Sign 0.4550.890i-0.455 - 0.890i
Analytic cond. 14.750414.7504
Root an. cond. 3.840633.84063
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  + (−0.618 − 1.90i)2-s + (−6.71 + 4.88i)3-s + (−3.23 + 2.35i)4-s + (13.4 + 9.76i)6-s + 11.9·7-s + (6.47 + 4.70i)8-s + (12.9 − 39.9i)9-s + (16.7 + 51.6i)11-s + (10.2 − 31.5i)12-s + (20.9 − 64.4i)13-s + (−7.39 − 22.7i)14-s + (4.94 − 15.2i)16-s + (−8.59 − 6.24i)17-s − 83.9·18-s + (−18.5 − 13.5i)19-s + ⋯
L(s)  = 1  + (−0.218 − 0.672i)2-s + (−1.29 + 0.939i)3-s + (−0.404 + 0.293i)4-s + (0.914 + 0.664i)6-s + 0.645·7-s + (0.286 + 0.207i)8-s + (0.480 − 1.47i)9-s + (0.460 + 1.41i)11-s + (0.246 − 0.760i)12-s + (0.447 − 1.37i)13-s + (−0.141 − 0.434i)14-s + (0.0772 − 0.237i)16-s + (−0.122 − 0.0891i)17-s − 1.09·18-s + (−0.224 − 0.163i)19-s + ⋯

Functional equation

Λ(s)=(250s/2ΓC(s)L(s)=((0.4550.890i)Λ(4s)\begin{aligned}\Lambda(s)=\mathstrut & 250 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.455 - 0.890i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(250s/2ΓC(s+3/2)L(s)=((0.4550.890i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 250 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.455 - 0.890i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 250250    =    2532 \cdot 5^{3}
Sign: 0.4550.890i-0.455 - 0.890i
Analytic conductor: 14.750414.7504
Root analytic conductor: 3.840633.84063
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ250(151,)\chi_{250} (151, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 250, ( :3/2), 0.4550.890i)(2,\ 250,\ (\ :3/2),\ -0.455 - 0.890i)

Particular Values

L(2)L(2) \approx 0.316107+0.517109i0.316107 + 0.517109i
L(12)L(\frac12) \approx 0.316107+0.517109i0.316107 + 0.517109i
L(52)L(\frac{5}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=pFp(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1+(0.618+1.90i)T 1 + (0.618 + 1.90i)T
5 1 1
good3 1+(6.714.88i)T+(8.3425.6i)T2 1 + (6.71 - 4.88i)T + (8.34 - 25.6i)T^{2}
7 111.9T+343T2 1 - 11.9T + 343T^{2}
11 1+(16.751.6i)T+(1.07e3+782.i)T2 1 + (-16.7 - 51.6i)T + (-1.07e3 + 782. i)T^{2}
13 1+(20.9+64.4i)T+(1.77e31.29e3i)T2 1 + (-20.9 + 64.4i)T + (-1.77e3 - 1.29e3i)T^{2}
17 1+(8.59+6.24i)T+(1.51e3+4.67e3i)T2 1 + (8.59 + 6.24i)T + (1.51e3 + 4.67e3i)T^{2}
19 1+(18.5+13.5i)T+(2.11e3+6.52e3i)T2 1 + (18.5 + 13.5i)T + (2.11e3 + 6.52e3i)T^{2}
23 1+(45.4140.i)T+(9.84e3+7.15e3i)T2 1 + (-45.4 - 140. i)T + (-9.84e3 + 7.15e3i)T^{2}
29 1+(166.121.i)T+(7.53e32.31e4i)T2 1 + (166. - 121. i)T + (7.53e3 - 2.31e4i)T^{2}
31 1+(70.951.5i)T+(9.20e3+2.83e4i)T2 1 + (-70.9 - 51.5i)T + (9.20e3 + 2.83e4i)T^{2}
37 1+(127.393.i)T+(4.09e42.97e4i)T2 1 + (127. - 393. i)T + (-4.09e4 - 2.97e4i)T^{2}
41 1+(20.964.3i)T+(5.57e44.05e4i)T2 1 + (20.9 - 64.3i)T + (-5.57e4 - 4.05e4i)T^{2}
43 1+471.T+7.95e4T2 1 + 471.T + 7.95e4T^{2}
47 1+(53.4+38.8i)T+(3.20e49.87e4i)T2 1 + (-53.4 + 38.8i)T + (3.20e4 - 9.87e4i)T^{2}
53 1+(255.+185.i)T+(4.60e41.41e5i)T2 1 + (-255. + 185. i)T + (4.60e4 - 1.41e5i)T^{2}
59 1+(31.396.3i)T+(1.66e51.20e5i)T2 1 + (31.3 - 96.3i)T + (-1.66e5 - 1.20e5i)T^{2}
61 1+(125.+385.i)T+(1.83e5+1.33e5i)T2 1 + (125. + 385. i)T + (-1.83e5 + 1.33e5i)T^{2}
67 1+(425.+308.i)T+(9.29e4+2.86e5i)T2 1 + (425. + 308. i)T + (9.29e4 + 2.86e5i)T^{2}
71 1+(386.+280.i)T+(1.10e53.40e5i)T2 1 + (-386. + 280. i)T + (1.10e5 - 3.40e5i)T^{2}
73 1+(264.815.i)T+(3.14e5+2.28e5i)T2 1 + (-264. - 815. i)T + (-3.14e5 + 2.28e5i)T^{2}
79 1+(444.322.i)T+(1.52e54.68e5i)T2 1 + (444. - 322. i)T + (1.52e5 - 4.68e5i)T^{2}
83 1+(232.169.i)T+(1.76e5+5.43e5i)T2 1 + (-232. - 169. i)T + (1.76e5 + 5.43e5i)T^{2}
89 1+(12.3+37.9i)T+(5.70e5+4.14e5i)T2 1 + (12.3 + 37.9i)T + (-5.70e5 + 4.14e5i)T^{2}
97 1+(833.605.i)T+(2.82e58.68e5i)T2 1 + (833. - 605. i)T + (2.82e5 - 8.68e5i)T^{2}
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   L(s)=p j=12(1αj,pps)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

−11.65881636332235965650743175777, −10.99260727311940229972259452559, −10.17077617291388288120641946119, −9.520478843970382410709517155349, −8.182649435778756152928366386462, −6.84099289432263308206392499347, −5.35727673889344020993095979882, −4.75904459517460167934848188134, −3.50474614082634548632521361415, −1.42015569879401814130074604148, 0.33819275537822039371571172382, 1.65118809403838743887997952946, 4.22043468190699007724740408254, 5.53480154067957260945344033925, 6.29917445547807911518291997972, 7.02392613189338579829953290987, 8.216812338144935410800777692422, 9.074172134806216855438541010358, 10.72216828499353197186171295261, 11.33176309870465686291007345521

Graph of the ZZ-function along the critical line