Properties

Label 1-920-920.29-r0-0-0
Degree 11
Conductor 920920
Sign 0.635+0.771i0.635 + 0.771i
Analytic cond. 4.272464.27246
Root an. cond. 4.272464.27246
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

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

Λ(s)=(920s/2ΓR(s)L(s)=((0.635+0.771i)Λ(1s)\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}
Λ(s)=(920s/2ΓR(s)L(s)=((0.635+0.771i)Λ(1s)\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: 11
Conductor: 920920    =    235232^{3} \cdot 5 \cdot 23
Sign: 0.635+0.771i0.635 + 0.771i
Analytic conductor: 4.272464.27246
Root analytic conductor: 4.272464.27246
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ920(29,)\chi_{920} (29, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (1, 920, (0: ), 0.635+0.771i)(1,\ 920,\ (0:\ ),\ 0.635 + 0.771i)

Particular Values

L(12)L(\frac{1}{2}) \approx 2.087358173+0.9850567605i2.087358173 + 0.9850567605i
L(12)L(\frac12) \approx 2.087358173+0.9850567605i2.087358173 + 0.9850567605i
L(1)L(1) \approx 1.528708790+0.3863474512i1.528708790 + 0.3863474512i
L(1)L(1) \approx 1.528708790+0.3863474512i1.528708790 + 0.3863474512i

Euler product

   L(s)=pFp(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1 1
5 1 1
23 1 1
good3 1+(0.841+0.540i)T 1 + (0.841 + 0.540i)T
7 1+(0.959+0.281i)T 1 + (0.959 + 0.281i)T
11 1+(0.6540.755i)T 1 + (0.654 - 0.755i)T
13 1+(0.959+0.281i)T 1 + (-0.959 + 0.281i)T
17 1+(0.142+0.989i)T 1 + (0.142 + 0.989i)T
19 1+(0.1420.989i)T 1 + (0.142 - 0.989i)T
29 1+(0.142+0.989i)T 1 + (0.142 + 0.989i)T
31 1+(0.8410.540i)T 1 + (0.841 - 0.540i)T
37 1+(0.415+0.909i)T 1 + (0.415 + 0.909i)T
41 1+(0.4150.909i)T 1 + (0.415 - 0.909i)T
43 1+(0.841+0.540i)T 1 + (0.841 + 0.540i)T
47 1T 1 - T
53 1+(0.9590.281i)T 1 + (-0.959 - 0.281i)T
59 1+(0.9590.281i)T 1 + (0.959 - 0.281i)T
61 1+(0.841+0.540i)T 1 + (-0.841 + 0.540i)T
67 1+(0.6540.755i)T 1 + (-0.654 - 0.755i)T
71 1+(0.6540.755i)T 1 + (-0.654 - 0.755i)T
73 1+(0.1420.989i)T 1 + (0.142 - 0.989i)T
79 1+(0.959+0.281i)T 1 + (-0.959 + 0.281i)T
83 1+(0.415+0.909i)T 1 + (0.415 + 0.909i)T
89 1+(0.841+0.540i)T 1 + (0.841 + 0.540i)T
97 1+(0.415+0.909i)T 1 + (-0.415 + 0.909i)T
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   L(s)=p (1αpps)1L(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 ZZ-function along the critical line