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Properties

Label 2-500-100.23-c1-0-24
Degree 22
Conductor 500500
Sign 0.967+0.251i0.967 + 0.251i
Analytic cond. 3.992523.99252
Root an. cond. 1.998121.99812
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.41 + 0.0976i)2-s + (−0.0867 − 0.170i)3-s + (1.98 + 0.275i)4-s + (−0.105 − 0.248i)6-s + (−1.89 − 1.89i)7-s + (2.76 + 0.582i)8-s + (1.74 − 2.39i)9-s + (2.99 + 4.12i)11-s + (−0.124 − 0.361i)12-s + (−0.433 − 2.73i)13-s + (−2.48 − 2.85i)14-s + (3.84 + 1.09i)16-s + (4.16 + 2.12i)17-s + (2.69 − 3.21i)18-s + (−1.55 − 4.77i)19-s + ⋯
L(s)  = 1  + (0.997 + 0.0690i)2-s + (−0.0500 − 0.0982i)3-s + (0.990 + 0.137i)4-s + (−0.0431 − 0.101i)6-s + (−0.715 − 0.715i)7-s + (0.978 + 0.205i)8-s + (0.580 − 0.799i)9-s + (0.904 + 1.24i)11-s + (−0.0360 − 0.104i)12-s + (−0.120 − 0.758i)13-s + (−0.664 − 0.763i)14-s + (0.962 + 0.273i)16-s + (1.01 + 0.514i)17-s + (0.634 − 0.757i)18-s + (−0.355 − 1.09i)19-s + ⋯

Functional equation

Λ(s)=(500s/2 ΓC(s) L(s)=((0.967+0.251i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.967 + 0.251i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(500s/2 ΓC(s+1/2) L(s)=((0.967+0.251i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 500 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.967 + 0.251i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 500500    =    22⋅532^{2} \cdot 5^{3}
Sign: 0.967+0.251i0.967 + 0.251i
Analytic conductor: 3.992523.99252
Root analytic conductor: 1.998121.99812
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ500(343,⋅)\chi_{500} (343, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 500, ( :1/2), 0.967+0.251i)(2,\ 500,\ (\ :1/2),\ 0.967 + 0.251i)

Particular Values

L(1)L(1) ≈\approx 2.62323−0.335890i2.62323 - 0.335890i
L(12)L(\frac12) ≈\approx 2.62323−0.335890i2.62323 - 0.335890i
L(32)L(\frac{3}{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+(−1.41−0.0976i)T 1 + (-1.41 - 0.0976i)T
5 1 1
good3 1+(0.0867+0.170i)T+(−1.76+2.42i)T2 1 + (0.0867 + 0.170i)T + (-1.76 + 2.42i)T^{2}
7 1+(1.89+1.89i)T+7iT2 1 + (1.89 + 1.89i)T + 7iT^{2}
11 1+(−2.99−4.12i)T+(−3.39+10.4i)T2 1 + (-2.99 - 4.12i)T + (-3.39 + 10.4i)T^{2}
13 1+(0.433+2.73i)T+(−12.3+4.01i)T2 1 + (0.433 + 2.73i)T + (-12.3 + 4.01i)T^{2}
17 1+(−4.16−2.12i)T+(9.99+13.7i)T2 1 + (-4.16 - 2.12i)T + (9.99 + 13.7i)T^{2}
19 1+(1.55+4.77i)T+(−15.3+11.1i)T2 1 + (1.55 + 4.77i)T + (-15.3 + 11.1i)T^{2}
23 1+(0.699−4.41i)T+(−21.8−7.10i)T2 1 + (0.699 - 4.41i)T + (-21.8 - 7.10i)T^{2}
29 1+(0.211+0.0688i)T+(23.4+17.0i)T2 1 + (0.211 + 0.0688i)T + (23.4 + 17.0i)T^{2}
31 1+(7.01−2.28i)T+(25.0−18.2i)T2 1 + (7.01 - 2.28i)T + (25.0 - 18.2i)T^{2}
37 1+(4.32−0.684i)T+(35.1−11.4i)T2 1 + (4.32 - 0.684i)T + (35.1 - 11.4i)T^{2}
41 1+(−2.54−1.84i)T+(12.6+38.9i)T2 1 + (-2.54 - 1.84i)T + (12.6 + 38.9i)T^{2}
43 1+(2.68−2.68i)T−43iT2 1 + (2.68 - 2.68i)T - 43iT^{2}
47 1+(7.84−3.99i)T+(27.6−38.0i)T2 1 + (7.84 - 3.99i)T + (27.6 - 38.0i)T^{2}
53 1+(1.64−0.837i)T+(31.1−42.8i)T2 1 + (1.64 - 0.837i)T + (31.1 - 42.8i)T^{2}
59 1+(7.33+5.32i)T+(18.2+56.1i)T2 1 + (7.33 + 5.32i)T + (18.2 + 56.1i)T^{2}
61 1+(−1.16+0.845i)T+(18.8−58.0i)T2 1 + (-1.16 + 0.845i)T + (18.8 - 58.0i)T^{2}
67 1+(−1.53+3.01i)T+(−39.3−54.2i)T2 1 + (-1.53 + 3.01i)T + (-39.3 - 54.2i)T^{2}
71 1+(−5.45−1.77i)T+(57.4+41.7i)T2 1 + (-5.45 - 1.77i)T + (57.4 + 41.7i)T^{2}
73 1+(−0.0187−0.00296i)T+(69.4+22.5i)T2 1 + (-0.0187 - 0.00296i)T + (69.4 + 22.5i)T^{2}
79 1+(2.08−6.41i)T+(−63.9−46.4i)T2 1 + (2.08 - 6.41i)T + (-63.9 - 46.4i)T^{2}
83 1+(−2.99−1.52i)T+(48.7+67.1i)T2 1 + (-2.99 - 1.52i)T + (48.7 + 67.1i)T^{2}
89 1+(0.509+0.700i)T+(−27.5+84.6i)T2 1 + (0.509 + 0.700i)T + (-27.5 + 84.6i)T^{2}
97 1+(−5.07−9.96i)T+(−57.0+78.4i)T2 1 + (-5.07 - 9.96i)T + (-57.0 + 78.4i)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

−11.02073733343802879675000761828, −10.03362160292075504437894463598, −9.426331171121367941107677441457, −7.75913437043501296369658847803, −6.97558150305228051088046247891, −6.40360198681666219869829747168, −5.13476314987413108521421694551, −4.00676303401685542302663530969, −3.29896411803659178156042560792, −1.50650807081196552884007559276, 1.82684550958868510369280224282, 3.21543277945067803287943904788, 4.12763561896804587675045671493, 5.39738094779256202698561547962, 6.12985211576064188920360067785, 7.03191039722717076742862948718, 8.176436835979998165651746242901, 9.311503850396833092941243398887, 10.26488798750085413638515400897, 11.13982528039500833249634181365

Graph of the ZZ-function along the critical line