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Properties

Label 1-20e2-400.91-r1-0-0
Degree $1$
Conductor $400$
Sign $0.109 - 0.993i$
Analytic cond. $42.9859$
Root an. cond. $42.9859$
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  + (−0.587 − 0.809i)3-s + 7-s + (−0.309 + 0.951i)9-s + (0.951 − 0.309i)11-s + (−0.951 − 0.309i)13-s + (−0.809 − 0.587i)17-s + (0.587 − 0.809i)19-s + (−0.587 − 0.809i)21-s + (0.309 + 0.951i)23-s + (0.951 − 0.309i)27-s + (0.587 + 0.809i)29-s + (0.809 + 0.587i)31-s + (−0.809 − 0.587i)33-s + (0.951 + 0.309i)37-s + (0.309 + 0.951i)39-s + ⋯
L(s)  = 1  + (−0.587 − 0.809i)3-s + 7-s + (−0.309 + 0.951i)9-s + (0.951 − 0.309i)11-s + (−0.951 − 0.309i)13-s + (−0.809 − 0.587i)17-s + (0.587 − 0.809i)19-s + (−0.587 − 0.809i)21-s + (0.309 + 0.951i)23-s + (0.951 − 0.309i)27-s + (0.587 + 0.809i)29-s + (0.809 + 0.587i)31-s + (−0.809 − 0.587i)33-s + (0.951 + 0.309i)37-s + (0.309 + 0.951i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.109 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.109 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(400\)    =    \(2^{4} \cdot 5^{2}\)
Sign: $0.109 - 0.993i$
Analytic conductor: \(42.9859\)
Root analytic conductor: \(42.9859\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{400} (91, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 400,\ (1:\ ),\ 0.109 - 0.993i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.293210445 - 1.158295888i\)
\(L(\frac12)\) \(\approx\) \(1.293210445 - 1.158295888i\)
\(L(1)\) \(\approx\) \(0.9829039062 - 0.3405582920i\)
\(L(1)\) \(\approx\) \(0.9829039062 - 0.3405582920i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
good3 \( 1 + (-0.587 - 0.809i)T \)
7 \( 1 + T \)
11 \( 1 + (0.951 - 0.309i)T \)
13 \( 1 + (-0.951 - 0.309i)T \)
17 \( 1 + (-0.809 - 0.587i)T \)
19 \( 1 + (0.587 - 0.809i)T \)
23 \( 1 + (0.309 + 0.951i)T \)
29 \( 1 + (0.587 + 0.809i)T \)
31 \( 1 + (0.809 + 0.587i)T \)
37 \( 1 + (0.951 + 0.309i)T \)
41 \( 1 + (-0.309 + 0.951i)T \)
43 \( 1 - iT \)
47 \( 1 + (0.809 - 0.587i)T \)
53 \( 1 + (-0.587 - 0.809i)T \)
59 \( 1 + (-0.951 - 0.309i)T \)
61 \( 1 + (0.951 - 0.309i)T \)
67 \( 1 + (0.587 - 0.809i)T \)
71 \( 1 + (-0.809 + 0.587i)T \)
73 \( 1 + (-0.309 - 0.951i)T \)
79 \( 1 + (0.809 - 0.587i)T \)
83 \( 1 + (0.587 - 0.809i)T \)
89 \( 1 + (-0.309 - 0.951i)T \)
97 \( 1 + (-0.809 + 0.587i)T \)
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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

−24.40429174589260530942732011066, −23.41335399917547721548981180696, −22.43521599038844407978541090499, −21.90348834910131524277364092633, −20.93281243481760153907047801550, −20.267870034800607225111933765129, −19.18690605904196267086997833368, −17.94880353281609300565855723269, −17.250658705523549453076653781788, −16.67512545394165358404876070566, −15.45669856901953140663680428188, −14.726560761399761383338417120660, −14.05037172473448975632986711288, −12.4260735239730169657860298176, −11.76197978903919642673163791170, −10.93850345033945104830599710961, −9.95928944819679448323125994315, −9.10864515803497362274110870676, −8.02185486507066350635501613249, −6.75267069212174285283378879271, −5.77711194264764628067355814907, −4.55515308929028769296972881548, −4.15521045116038636062100274244, −2.45884923624239549342618073066, −1.02818370017051147559352811042, 0.63637228535188231344127335856, 1.676186498335017655446759208927, 2.89530231194757666247066766648, 4.645799210995743511562130632472, 5.28730328971361129440861444284, 6.60316611251755989691718198589, 7.31972862174457435253500707681, 8.316030520429031050395371018746, 9.38075681507452382086884302189, 10.75333856331564520105943776158, 11.58482380227898414729153531166, 12.08346855714377634968564490797, 13.35885212415203562168668666764, 14.04635065297367282871046662805, 15.03271326489314152615988973349, 16.193328679955681895102050791534, 17.39115853534394990918264050910, 17.57189123121780110647919514362, 18.60969927698083590695558394329, 19.66061314539777838274975212670, 20.21110952138685260364115854431, 21.76649415683752932274669922060, 22.08167205753726244897994154214, 23.26161609272347060730483039794, 24.026875697460050692050717823445

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