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Properties

Label 1-40e2-1600.869-r0-0-0
Degree $1$
Conductor $1600$
Sign $0.704 + 0.709i$
Analytic cond. $7.43036$
Root an. cond. $7.43036$
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  + (0.996 − 0.0784i)3-s + (0.707 + 0.707i)7-s + (0.987 − 0.156i)9-s + (0.233 + 0.972i)11-s + (−0.972 − 0.233i)13-s + (−0.951 + 0.309i)17-s + (0.649 + 0.760i)19-s + (0.760 + 0.649i)21-s + (0.156 − 0.987i)23-s + (0.972 − 0.233i)27-s + (0.996 − 0.0784i)29-s + (−0.309 − 0.951i)31-s + (0.309 + 0.951i)33-s + (0.522 + 0.852i)37-s + (−0.987 − 0.156i)39-s + ⋯
L(s)  = 1  + (0.996 − 0.0784i)3-s + (0.707 + 0.707i)7-s + (0.987 − 0.156i)9-s + (0.233 + 0.972i)11-s + (−0.972 − 0.233i)13-s + (−0.951 + 0.309i)17-s + (0.649 + 0.760i)19-s + (0.760 + 0.649i)21-s + (0.156 − 0.987i)23-s + (0.972 − 0.233i)27-s + (0.996 − 0.0784i)29-s + (−0.309 − 0.951i)31-s + (0.309 + 0.951i)33-s + (0.522 + 0.852i)37-s + (−0.987 − 0.156i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.704 + 0.709i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.704 + 0.709i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1600\)    =    \(2^{6} \cdot 5^{2}\)
Sign: $0.704 + 0.709i$
Analytic conductor: \(7.43036\)
Root analytic conductor: \(7.43036\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1600} (869, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1600,\ (0:\ ),\ 0.704 + 0.709i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.319870065 + 0.9662625727i\)
\(L(\frac12)\) \(\approx\) \(2.319870065 + 0.9662625727i\)
\(L(1)\) \(\approx\) \(1.585229352 + 0.2508517429i\)
\(L(1)\) \(\approx\) \(1.585229352 + 0.2508517429i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
good3 \( 1 + (0.996 - 0.0784i)T \)
7 \( 1 + (0.707 + 0.707i)T \)
11 \( 1 + (0.233 + 0.972i)T \)
13 \( 1 + (-0.972 - 0.233i)T \)
17 \( 1 + (-0.951 + 0.309i)T \)
19 \( 1 + (0.649 + 0.760i)T \)
23 \( 1 + (0.156 - 0.987i)T \)
29 \( 1 + (0.996 - 0.0784i)T \)
31 \( 1 + (-0.309 - 0.951i)T \)
37 \( 1 + (0.522 + 0.852i)T \)
41 \( 1 + (-0.987 + 0.156i)T \)
43 \( 1 + (0.382 - 0.923i)T \)
47 \( 1 + (0.951 + 0.309i)T \)
53 \( 1 + (0.760 + 0.649i)T \)
59 \( 1 + (0.522 + 0.852i)T \)
61 \( 1 + (0.852 + 0.522i)T \)
67 \( 1 + (-0.760 + 0.649i)T \)
71 \( 1 + (-0.891 + 0.453i)T \)
73 \( 1 + (-0.156 + 0.987i)T \)
79 \( 1 + (0.951 + 0.309i)T \)
83 \( 1 + (-0.649 - 0.760i)T \)
89 \( 1 + (-0.987 - 0.156i)T \)
97 \( 1 + (0.309 - 0.951i)T \)
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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

−20.17524072993830385261672928186, −19.71570480764590976902773632849, −19.20527731868030991945411370862, −18.03989842997714062948921183473, −17.58334652849679593786192548107, −16.53240541849848902220325731379, −15.85507975319824953294797193120, −15.02271235295940139684384359899, −14.24390552592996482220196475771, −13.75929178202580047005313099607, −13.160999496729999523708303446297, −12.003526662828225052518681101687, −11.21081333999862531443323382883, −10.46493042808893756020379937346, −9.50275482404888944593884188482, −8.88780236461145462545578896251, −8.08293857135959137624942735532, −7.27972547758179292955124896145, −6.73006018279585195221410486235, −5.24975510296305154692332061886, −4.582257213110832823035154195623, −3.67770314602418357117734439412, −2.84398715364720510886653851812, −1.91084963940284504113123151275, −0.8557717324649530491943788948, 1.30677070282342487921364910480, 2.292130880654828484984784281700, 2.70260537239975175449892381115, 4.129429201241283855709392288620, 4.6204419211393376939328444212, 5.67856155682794103010248729559, 6.85402022035633328300701079338, 7.50164535692741658393349685119, 8.358889556755827578842552100904, 8.926915175403085586479412636746, 9.850716514877937827308946223699, 10.40938225338997954974575486720, 11.73664695223565087544276777073, 12.2645922745282840916477130178, 13.03354978730046300869471176603, 13.94979780840102729627025676608, 14.76388654188824681189815670848, 15.05319638790630992490313572216, 15.81584809386735679705600011846, 16.95393485995451706578482913069, 17.72733636834358695023664207957, 18.425075437413953418005981221938, 19.05662979874459444881742308440, 20.063738292172068653887874550715, 20.37263094555320035600776609090

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