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Properties

Label 2-500-100.67-c1-0-6
Degree $2$
Conductor $500$
Sign $0.586 - 0.810i$
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  + (−0.238 − 1.39i)2-s + (0.745 + 0.118i)3-s + (−1.88 + 0.664i)4-s + (−0.0129 − 1.06i)6-s + (−2.75 + 2.75i)7-s + (1.37 + 2.47i)8-s + (−2.31 − 0.750i)9-s + (0.712 − 0.231i)11-s + (−1.48 + 0.272i)12-s + (0.753 + 1.47i)13-s + (4.49 + 3.18i)14-s + (3.11 − 2.50i)16-s + (0.715 + 4.51i)17-s + (−0.496 + 3.40i)18-s + (2.96 + 2.15i)19-s + ⋯
L(s)  = 1  + (−0.168 − 0.985i)2-s + (0.430 + 0.0681i)3-s + (−0.943 + 0.332i)4-s + (−0.00529 − 0.435i)6-s + (−1.04 + 1.04i)7-s + (0.486 + 0.873i)8-s + (−0.770 − 0.250i)9-s + (0.214 − 0.0698i)11-s + (−0.428 + 0.0786i)12-s + (0.208 + 0.409i)13-s + (1.20 + 0.851i)14-s + (0.779 − 0.626i)16-s + (0.173 + 1.09i)17-s + (−0.116 + 0.801i)18-s + (0.679 + 0.493i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.586 - 0.810i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 500 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.586 - 0.810i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(500\)    =    \(2^{2} \cdot 5^{3}\)
Sign: $0.586 - 0.810i$
Analytic conductor: \(3.99252\)
Root analytic conductor: \(1.99812\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{500} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 500,\ (\ :1/2),\ 0.586 - 0.810i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.721931 + 0.368607i\)
\(L(\frac12)\) \(\approx\) \(0.721931 + 0.368607i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.238 + 1.39i)T \)
5 \( 1 \)
good3 \( 1 + (-0.745 - 0.118i)T + (2.85 + 0.927i)T^{2} \)
7 \( 1 + (2.75 - 2.75i)T - 7iT^{2} \)
11 \( 1 + (-0.712 + 0.231i)T + (8.89 - 6.46i)T^{2} \)
13 \( 1 + (-0.753 - 1.47i)T + (-7.64 + 10.5i)T^{2} \)
17 \( 1 + (-0.715 - 4.51i)T + (-16.1 + 5.25i)T^{2} \)
19 \( 1 + (-2.96 - 2.15i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (4.00 - 7.86i)T + (-13.5 - 18.6i)T^{2} \)
29 \( 1 + (0.00370 + 0.00510i)T + (-8.96 + 27.5i)T^{2} \)
31 \( 1 + (1.08 - 1.49i)T + (-9.57 - 29.4i)T^{2} \)
37 \( 1 + (-3.22 + 1.64i)T + (21.7 - 29.9i)T^{2} \)
41 \( 1 + (1.93 - 5.94i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + (6.10 + 6.10i)T + 43iT^{2} \)
47 \( 1 + (0.184 - 1.16i)T + (-44.6 - 14.5i)T^{2} \)
53 \( 1 + (1.44 - 9.11i)T + (-50.4 - 16.3i)T^{2} \)
59 \( 1 + (-4.58 + 14.0i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (0.569 + 1.75i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + (-5.08 + 0.805i)T + (63.7 - 20.7i)T^{2} \)
71 \( 1 + (6.26 + 8.62i)T + (-21.9 + 67.5i)T^{2} \)
73 \( 1 + (-6.26 - 3.19i)T + (42.9 + 59.0i)T^{2} \)
79 \( 1 + (2.90 - 2.11i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (0.308 + 1.94i)T + (-78.9 + 25.6i)T^{2} \)
89 \( 1 + (3.79 - 1.23i)T + (72.0 - 52.3i)T^{2} \)
97 \( 1 + (-6.43 - 1.01i)T + (92.2 + 29.9i)T^{2} \)
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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

−11.21441743377407366498113464064, −9.930587803987360953668319246174, −9.431072173015429863983174297541, −8.675545643864835428741627710606, −7.86584174277527097507346474475, −6.21960432693942991911892032612, −5.45003315119071807841494724625, −3.74733781865765515640259287212, −3.16200292333112962342604059123, −1.86969335683013801937489181838, 0.48344675034558058831449154951, 2.95505927771991863097305314718, 4.07883511079919041905869037981, 5.29220208744793899137617507133, 6.39407738930239879257249932840, 7.14183767393947321209351212711, 8.008300199884435879217671986231, 8.890529450864693240032473999008, 9.772512618599810984005299724614, 10.41958949109364727498786750637

Graph of the $Z$-function along the critical line