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Properties

Label 1-920-920.29-r0-0-0
Degree $1$
Conductor $920$
Sign $0.635 + 0.771i$
Analytic cond. $4.27246$
Root an. cond. $4.27246$
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  + (0.841 + 0.540i)3-s + (0.959 + 0.281i)7-s + (0.415 + 0.909i)9-s + (0.654 − 0.755i)11-s + (−0.959 + 0.281i)13-s + (0.142 + 0.989i)17-s + (0.142 − 0.989i)19-s + (0.654 + 0.755i)21-s + (−0.142 + 0.989i)27-s + (0.142 + 0.989i)29-s + (0.841 − 0.540i)31-s + (0.959 − 0.281i)33-s + (0.415 + 0.909i)37-s + (−0.959 − 0.281i)39-s + (0.415 − 0.909i)41-s + ⋯
L(s)  = 1  + (0.841 + 0.540i)3-s + (0.959 + 0.281i)7-s + (0.415 + 0.909i)9-s + (0.654 − 0.755i)11-s + (−0.959 + 0.281i)13-s + (0.142 + 0.989i)17-s + (0.142 − 0.989i)19-s + (0.654 + 0.755i)21-s + (−0.142 + 0.989i)27-s + (0.142 + 0.989i)29-s + (0.841 − 0.540i)31-s + (0.959 − 0.281i)33-s + (0.415 + 0.909i)37-s + (−0.959 − 0.281i)39-s + (0.415 − 0.909i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.635 + 0.771i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.635 + 0.771i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(920\)    =    \(2^{3} \cdot 5 \cdot 23\)
Sign: $0.635 + 0.771i$
Analytic conductor: \(4.27246\)
Root analytic conductor: \(4.27246\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{920} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 920,\ (0:\ ),\ 0.635 + 0.771i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.087358173 + 0.9850567605i\)
\(L(\frac12)\) \(\approx\) \(2.087358173 + 0.9850567605i\)
\(L(1)\) \(\approx\) \(1.528708790 + 0.3863474512i\)
\(L(1)\) \(\approx\) \(1.528708790 + 0.3863474512i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
23 \( 1 \)
good3 \( 1 + (0.841 + 0.540i)T \)
7 \( 1 + (0.959 + 0.281i)T \)
11 \( 1 + (0.654 - 0.755i)T \)
13 \( 1 + (-0.959 + 0.281i)T \)
17 \( 1 + (0.142 + 0.989i)T \)
19 \( 1 + (0.142 - 0.989i)T \)
29 \( 1 + (0.142 + 0.989i)T \)
31 \( 1 + (0.841 - 0.540i)T \)
37 \( 1 + (0.415 + 0.909i)T \)
41 \( 1 + (0.415 - 0.909i)T \)
43 \( 1 + (0.841 + 0.540i)T \)
47 \( 1 - T \)
53 \( 1 + (-0.959 - 0.281i)T \)
59 \( 1 + (0.959 - 0.281i)T \)
61 \( 1 + (-0.841 + 0.540i)T \)
67 \( 1 + (-0.654 - 0.755i)T \)
71 \( 1 + (-0.654 - 0.755i)T \)
73 \( 1 + (0.142 - 0.989i)T \)
79 \( 1 + (-0.959 + 0.281i)T \)
83 \( 1 + (0.415 + 0.909i)T \)
89 \( 1 + (0.841 + 0.540i)T \)
97 \( 1 + (-0.415 + 0.909i)T \)
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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

−21.558520284373004111210404590269, −20.7751353391280672645232472593, −20.23078702876610283768723397817, −19.49281643181592478913283280939, −18.67934204077557038900158181147, −17.728500278262349235835489212203, −17.373436511447444959715471193390, −16.15047119838922199358249674739, −15.101228628884374293638842948782, −14.42153652776897388947578597351, −14.0557907313109038423958816641, −12.93715654108195515753594644393, −12.112269973181270956075507614462, −11.53270581158297446006979938424, −10.13089547718020274707650315567, −9.55917958149019308526551110396, −8.54172728829168396008697938249, −7.62970378513755154548535874309, −7.2675320467051106129607813293, −6.09491108196236120919749881847, −4.81718092129310798422793972039, −4.09519037061779428741393713528, −2.86795605554387530816153055150, −1.99104954647287218252188057157, −1.04132034427300331374314339370, 1.36617228174320900524573103931, 2.37865442856570264536102922133, 3.303578316188690573237726371063, 4.39805265924032939252494038752, 4.99056161697838324829776770406, 6.19442059254755922025941629112, 7.37302922756997024369483299182, 8.19380748941047269704468349980, 8.87854483020825412682890447350, 9.61802310797944349537130188812, 10.66049190409409281569834469550, 11.36290732552191066750549399789, 12.30843774147590066894963541838, 13.39256221890587421628878481716, 14.207856725285278069437271322140, 14.756994435427869797817998472832, 15.40778599583018127346580503629, 16.435642418226846299464033927894, 17.170542758958772330146404420663, 18.018527630116882910640803546673, 19.278335491707554995519882120443, 19.42677038476604668101565737053, 20.50459094322522013262229066384, 21.24615587353164187535377444186, 21.86474801497870127934628243215

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