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Properties

Label 2-500-100.23-c1-0-24
Degree $2$
Conductor $500$
Sign $0.967 + 0.251i$
Analytic cond. $3.99252$
Root an. cond. $1.99812$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

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

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

Invariants

Degree: \(2\)
Conductor: \(500\)    =    \(2^{2} \cdot 5^{3}\)
Sign: $0.967 + 0.251i$
Analytic conductor: \(3.99252\)
Root analytic conductor: \(1.99812\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{500} (343, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 500,\ (\ :1/2),\ 0.967 + 0.251i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.62323 - 0.335890i\)
\(L(\frac12)\) \(\approx\) \(2.62323 - 0.335890i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.41 - 0.0976i)T \)
5 \( 1 \)
good3 \( 1 + (0.0867 + 0.170i)T + (-1.76 + 2.42i)T^{2} \)
7 \( 1 + (1.89 + 1.89i)T + 7iT^{2} \)
11 \( 1 + (-2.99 - 4.12i)T + (-3.39 + 10.4i)T^{2} \)
13 \( 1 + (0.433 + 2.73i)T + (-12.3 + 4.01i)T^{2} \)
17 \( 1 + (-4.16 - 2.12i)T + (9.99 + 13.7i)T^{2} \)
19 \( 1 + (1.55 + 4.77i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (0.699 - 4.41i)T + (-21.8 - 7.10i)T^{2} \)
29 \( 1 + (0.211 + 0.0688i)T + (23.4 + 17.0i)T^{2} \)
31 \( 1 + (7.01 - 2.28i)T + (25.0 - 18.2i)T^{2} \)
37 \( 1 + (4.32 - 0.684i)T + (35.1 - 11.4i)T^{2} \)
41 \( 1 + (-2.54 - 1.84i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + (2.68 - 2.68i)T - 43iT^{2} \)
47 \( 1 + (7.84 - 3.99i)T + (27.6 - 38.0i)T^{2} \)
53 \( 1 + (1.64 - 0.837i)T + (31.1 - 42.8i)T^{2} \)
59 \( 1 + (7.33 + 5.32i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-1.16 + 0.845i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (-1.53 + 3.01i)T + (-39.3 - 54.2i)T^{2} \)
71 \( 1 + (-5.45 - 1.77i)T + (57.4 + 41.7i)T^{2} \)
73 \( 1 + (-0.0187 - 0.00296i)T + (69.4 + 22.5i)T^{2} \)
79 \( 1 + (2.08 - 6.41i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-2.99 - 1.52i)T + (48.7 + 67.1i)T^{2} \)
89 \( 1 + (0.509 + 0.700i)T + (-27.5 + 84.6i)T^{2} \)
97 \( 1 + (-5.07 - 9.96i)T + (-57.0 + 78.4i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

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 $Z$-function along the critical line