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Properties

Label 1-500-500.71-r1-0-0
Degree $1$
Conductor $500$
Sign $-0.577 + 0.816i$
Analytic cond. $53.7324$
Root an. cond. $53.7324$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.968 − 0.248i)3-s + (−0.309 − 0.951i)7-s + (0.876 + 0.481i)9-s + (0.187 + 0.982i)11-s + (0.876 + 0.481i)13-s + (−0.929 − 0.368i)17-s + (−0.968 + 0.248i)19-s + (0.0627 + 0.998i)21-s + (0.425 − 0.904i)23-s + (−0.728 − 0.684i)27-s + (−0.637 − 0.770i)29-s + (0.929 + 0.368i)31-s + (0.0627 − 0.998i)33-s + (0.728 − 0.684i)37-s + (−0.728 − 0.684i)39-s + ⋯
L(s)  = 1  + (−0.968 − 0.248i)3-s + (−0.309 − 0.951i)7-s + (0.876 + 0.481i)9-s + (0.187 + 0.982i)11-s + (0.876 + 0.481i)13-s + (−0.929 − 0.368i)17-s + (−0.968 + 0.248i)19-s + (0.0627 + 0.998i)21-s + (0.425 − 0.904i)23-s + (−0.728 − 0.684i)27-s + (−0.637 − 0.770i)29-s + (0.929 + 0.368i)31-s + (0.0627 − 0.998i)33-s + (0.728 − 0.684i)37-s + (−0.728 − 0.684i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 500 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 500 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(500\)    =    \(2^{2} \cdot 5^{3}\)
Sign: $-0.577 + 0.816i$
Analytic conductor: \(53.7324\)
Root analytic conductor: \(53.7324\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{500} (71, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 500,\ (1:\ ),\ -0.577 + 0.816i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1392945787 + 0.2691862788i\)
\(L(\frac12)\) \(\approx\) \(0.1392945787 + 0.2691862788i\)
\(L(1)\) \(\approx\) \(0.6788872609 - 0.05728616751i\)
\(L(1)\) \(\approx\) \(0.6788872609 - 0.05728616751i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
good3 \( 1 + (-0.968 - 0.248i)T \)
7 \( 1 + (-0.309 - 0.951i)T \)
11 \( 1 + (0.187 + 0.982i)T \)
13 \( 1 + (0.876 + 0.481i)T \)
17 \( 1 + (-0.929 - 0.368i)T \)
19 \( 1 + (-0.968 + 0.248i)T \)
23 \( 1 + (0.425 - 0.904i)T \)
29 \( 1 + (-0.637 - 0.770i)T \)
31 \( 1 + (0.929 + 0.368i)T \)
37 \( 1 + (0.728 - 0.684i)T \)
41 \( 1 + (-0.425 - 0.904i)T \)
43 \( 1 + (0.809 - 0.587i)T \)
47 \( 1 + (-0.535 + 0.844i)T \)
53 \( 1 + (0.0627 + 0.998i)T \)
59 \( 1 + (0.992 - 0.125i)T \)
61 \( 1 + (-0.425 + 0.904i)T \)
67 \( 1 + (0.637 - 0.770i)T \)
71 \( 1 + (-0.535 + 0.844i)T \)
73 \( 1 + (-0.992 - 0.125i)T \)
79 \( 1 + (-0.968 - 0.248i)T \)
83 \( 1 + (-0.968 + 0.248i)T \)
89 \( 1 + (-0.992 - 0.125i)T \)
97 \( 1 + (-0.637 - 0.770i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−23.11362901524994526782222207512, −22.19240488435534090762889804539, −21.66232059468812924201319407704, −20.93718993449448922648865945845, −19.61270904264943939082039546061, −18.777535736979983725359062783364, −18.03926773480681185247058102104, −17.18110346266519950785978221522, −16.27557644100484004084754492346, −15.55857314116819692568478344734, −14.87405110991047275255886915119, −13.26311991997306359438019621463, −12.869010137699448519746492455198, −11.49804662964529048349156082200, −11.21449818024342282128457698434, −10.09374717888008075538385184179, −9.02572303514801611086029483313, −8.26480110652816135658726071215, −6.66537121031426394916213471540, −6.07440692634106300434120607841, −5.26281091458445630241235303672, −4.06936811600081130822335177523, −2.974066910613983857272837511093, −1.44609026893254781231228084761, −0.102539370977675588500762164285, 1.07588273458690703631923194723, 2.265472573609856031276235672481, 4.12078742422675438575104785007, 4.50839086439883146753578360142, 5.99853676695049233034080607216, 6.74450079749650270962514035736, 7.42780168940419960231018381014, 8.78054342383882518320632126586, 9.95094141184548252377386841122, 10.72382317893676208061162155890, 11.45135117956787630739248751656, 12.56068812612659616923590193266, 13.18588534816304458787212046172, 14.12901929530403811764259167014, 15.37541751853682623715513488791, 16.19619529981656906176216381260, 17.06419966547585582668235596158, 17.59392655308138444281159457150, 18.59480914087329785265138379505, 19.39293039240087358764872314430, 20.470770538825200310855464421002, 21.16299976014375735417544658890, 22.42263656380065305929487726579, 22.897276251181275904803731608921, 23.55212786892422816828739240774

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