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Properties

Label 1-500-500.119-r1-0-0
Degree $1$
Conductor $500$
Sign $0.0125 + 0.999i$
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  + (−0.929 − 0.368i)3-s + (0.309 − 0.951i)7-s + (0.728 + 0.684i)9-s + (−0.876 − 0.481i)11-s + (−0.728 − 0.684i)13-s + (−0.535 + 0.844i)17-s + (0.929 − 0.368i)19-s + (−0.637 + 0.770i)21-s + (−0.992 + 0.125i)23-s + (−0.425 − 0.904i)27-s + (0.968 − 0.248i)29-s + (−0.535 + 0.844i)31-s + (0.637 + 0.770i)33-s + (0.425 − 0.904i)37-s + (0.425 + 0.904i)39-s + ⋯
L(s)  = 1  + (−0.929 − 0.368i)3-s + (0.309 − 0.951i)7-s + (0.728 + 0.684i)9-s + (−0.876 − 0.481i)11-s + (−0.728 − 0.684i)13-s + (−0.535 + 0.844i)17-s + (0.929 − 0.368i)19-s + (−0.637 + 0.770i)21-s + (−0.992 + 0.125i)23-s + (−0.425 − 0.904i)27-s + (0.968 − 0.248i)29-s + (−0.535 + 0.844i)31-s + (0.637 + 0.770i)33-s + (0.425 − 0.904i)37-s + (0.425 + 0.904i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 500 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0125 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 500 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0125 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(500\)    =    \(2^{2} \cdot 5^{3}\)
Sign: $0.0125 + 0.999i$
Analytic conductor: \(53.7324\)
Root analytic conductor: \(53.7324\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{500} (119, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 500,\ (1:\ ),\ 0.0125 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1619347270 + 0.1639825526i\)
\(L(\frac12)\) \(\approx\) \(0.1619347270 + 0.1639825526i\)
\(L(1)\) \(\approx\) \(0.6337967861 - 0.1636767933i\)
\(L(1)\) \(\approx\) \(0.6337967861 - 0.1636767933i\)

Euler product

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

−23.19377288137426370746669622258, −22.20819768453338155056549170244, −21.79928266599081593696083425943, −20.84377827578521720860012741966, −20.032098881376400985448052131201, −18.45823979476906327397927568693, −18.336608522916844051956679622219, −17.31158503105787315532222210573, −16.291941740142639170955474868755, −15.663522979616601149824411031490, −14.86543677129155716995597226206, −13.738929464511170030693952427387, −12.49803961023928614236996911541, −11.89951673658049709152466709537, −11.18653645216397692543642464331, −9.95840503822073115824384569721, −9.45077243704619182570903254076, −8.11498426428559650891742192824, −7.0644478438381806757195464446, −6.02835580998541934510927124553, −5.07160538463966798189381582018, −4.50905547184652035564972929845, −2.905327533727909213943916630487, −1.75399750811716264666540532781, −0.082618005829870230002053958711, 0.88734935413712289476373285833, 2.19772618888710038134012294022, 3.67665972854307350738873062319, 4.88068329031101282654766184112, 5.57064675248524082351161302386, 6.75540544431627330089899538293, 7.55779171024435926609662045110, 8.36634460525743991001402908077, 10.09037459111948384237955644350, 10.49389382036630387312817430692, 11.43922170144830374603224151268, 12.36719310481007984996843213191, 13.28079019948588196530502474162, 13.92464217404136546028757097531, 15.24655427291570150653850141619, 16.13781842036366653654807042206, 16.91681147979187880250459061137, 17.81357796147706048007231120818, 18.2059489080470940314929530764, 19.5547157852918070307535285261, 20.09236873659090800971042502266, 21.37208846974602974337008962398, 21.99818854772673204309071313211, 22.9449977098045109588692912124, 23.73977980792632518995836407377

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