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Properties

Label 2-250-25.6-c3-0-17
Degree 22
Conductor 250250
Sign −0.836+0.547i-0.836 + 0.547i
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 + (4.17 − 3.03i)3-s + (−3.23 + 2.35i)4-s + (−8.34 − 6.06i)6-s + 3.36·7-s + (6.47 + 4.70i)8-s + (−0.124 + 0.383i)9-s + (−4.26 − 13.1i)11-s + (−6.37 + 19.6i)12-s + (28.1 − 86.7i)13-s + (−2.07 − 6.39i)14-s + (4.94 − 15.2i)16-s + (−44.3 − 32.2i)17-s + 0.806·18-s + (−49.0 − 35.6i)19-s + ⋯
L(s)  = 1  + (−0.218 − 0.672i)2-s + (0.802 − 0.583i)3-s + (−0.404 + 0.293i)4-s + (−0.567 − 0.412i)6-s + 0.181·7-s + (0.286 + 0.207i)8-s + (−0.00461 + 0.0142i)9-s + (−0.116 − 0.360i)11-s + (−0.153 + 0.471i)12-s + (0.601 − 1.85i)13-s + (−0.0396 − 0.122i)14-s + (0.0772 − 0.237i)16-s + (−0.632 − 0.459i)17-s + 0.0105·18-s + (−0.592 − 0.430i)19-s + ⋯

Functional equation

Λ(s)=(250s/2 ΓC(s) L(s)=((−0.836+0.547i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 250 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.836 + 0.547i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(250s/2 ΓC(s+3/2) L(s)=((−0.836+0.547i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 250 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.836 + 0.547i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 250250    =    2⋅532 \cdot 5^{3}
Sign: −0.836+0.547i-0.836 + 0.547i
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), −0.836+0.547i)(2,\ 250,\ (\ :3/2),\ -0.836 + 0.547i)

Particular Values

L(2)L(2) ≈\approx 0.483830−1.62373i0.483830 - 1.62373i
L(12)L(\frac12) ≈\approx 0.483830−1.62373i0.483830 - 1.62373i
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+(0.618+1.90i)T 1 + (0.618 + 1.90i)T
5 1 1
good3 1+(−4.17+3.03i)T+(8.34−25.6i)T2 1 + (-4.17 + 3.03i)T + (8.34 - 25.6i)T^{2}
7 1−3.36T+343T2 1 - 3.36T + 343T^{2}
11 1+(4.26+13.1i)T+(−1.07e3+782.i)T2 1 + (4.26 + 13.1i)T + (-1.07e3 + 782. i)T^{2}
13 1+(−28.1+86.7i)T+(−1.77e3−1.29e3i)T2 1 + (-28.1 + 86.7i)T + (-1.77e3 - 1.29e3i)T^{2}
17 1+(44.3+32.2i)T+(1.51e3+4.67e3i)T2 1 + (44.3 + 32.2i)T + (1.51e3 + 4.67e3i)T^{2}
19 1+(49.0+35.6i)T+(2.11e3+6.52e3i)T2 1 + (49.0 + 35.6i)T + (2.11e3 + 6.52e3i)T^{2}
23 1+(41.3+127.i)T+(−9.84e3+7.15e3i)T2 1 + (41.3 + 127. i)T + (-9.84e3 + 7.15e3i)T^{2}
29 1+(−125.+90.8i)T+(7.53e3−2.31e4i)T2 1 + (-125. + 90.8i)T + (7.53e3 - 2.31e4i)T^{2}
31 1+(77.5+56.3i)T+(9.20e3+2.83e4i)T2 1 + (77.5 + 56.3i)T + (9.20e3 + 2.83e4i)T^{2}
37 1+(4.12−12.6i)T+(−4.09e4−2.97e4i)T2 1 + (4.12 - 12.6i)T + (-4.09e4 - 2.97e4i)T^{2}
41 1+(47.9−147.i)T+(−5.57e4−4.05e4i)T2 1 + (47.9 - 147. i)T + (-5.57e4 - 4.05e4i)T^{2}
43 1−471.T+7.95e4T2 1 - 471.T + 7.95e4T^{2}
47 1+(145.−105.i)T+(3.20e4−9.87e4i)T2 1 + (145. - 105. i)T + (3.20e4 - 9.87e4i)T^{2}
53 1+(65.2−47.4i)T+(4.60e4−1.41e5i)T2 1 + (65.2 - 47.4i)T + (4.60e4 - 1.41e5i)T^{2}
59 1+(69.6−214.i)T+(−1.66e5−1.20e5i)T2 1 + (69.6 - 214. i)T + (-1.66e5 - 1.20e5i)T^{2}
61 1+(−90.5−278.i)T+(−1.83e5+1.33e5i)T2 1 + (-90.5 - 278. i)T + (-1.83e5 + 1.33e5i)T^{2}
67 1+(814.+591.i)T+(9.29e4+2.86e5i)T2 1 + (814. + 591. i)T + (9.29e4 + 2.86e5i)T^{2}
71 1+(−791.+575.i)T+(1.10e5−3.40e5i)T2 1 + (-791. + 575. i)T + (1.10e5 - 3.40e5i)T^{2}
73 1+(249.+769.i)T+(−3.14e5+2.28e5i)T2 1 + (249. + 769. i)T + (-3.14e5 + 2.28e5i)T^{2}
79 1+(−97.7+70.9i)T+(1.52e5−4.68e5i)T2 1 + (-97.7 + 70.9i)T + (1.52e5 - 4.68e5i)T^{2}
83 1+(−935.−679.i)T+(1.76e5+5.43e5i)T2 1 + (-935. - 679. i)T + (1.76e5 + 5.43e5i)T^{2}
89 1+(−399.−1.22e3i)T+(−5.70e5+4.14e5i)T2 1 + (-399. - 1.22e3i)T + (-5.70e5 + 4.14e5i)T^{2}
97 1+(−1.32e3+963.i)T+(2.82e5−8.68e5i)T2 1 + (-1.32e3 + 963. i)T + (2.82e5 - 8.68e5i)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

−11.00685304854704425937272519825, −10.49853912880352244068452204033, −9.130846387366896467617730469794, −8.269996337541394342146295307864, −7.72365656736590857277037428462, −6.19841366090155927774243778891, −4.74154166018124733874135339620, −3.18386039484593507762290045643, −2.29922538066669780121225329420, −0.65784281463471386596698839175, 1.85447032050923050413642730163, 3.72001115674016690901277046138, 4.55690487986235160866259051011, 6.10589223789264908146097166326, 7.06055636857564484734364668075, 8.338713523220947616682663902362, 8.998646210691774631558383460469, 9.725000048490753206849114197865, 10.86107940395757523006122373710, 11.95989718205363985966199469038

Graph of the ZZ-function along the critical line