Properties

Label 1-431-431.367-r1-0-0
Degree 11
Conductor 431431
Sign 0.114+0.993i-0.114 + 0.993i
Analytic cond. 46.317346.3173
Root an. cond. 46.317346.3173
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.581 + 0.813i)2-s + (−0.976 − 0.217i)3-s + (−0.322 − 0.946i)4-s + (0.639 − 0.768i)5-s + (0.744 − 0.667i)6-s + (−0.905 − 0.424i)7-s + (0.957 + 0.288i)8-s + (0.905 + 0.424i)9-s + (0.252 + 0.967i)10-s + (0.520 + 0.853i)11-s + (0.109 + 0.994i)12-s + (0.581 + 0.813i)13-s + (0.872 − 0.489i)14-s + (−0.791 + 0.611i)15-s + (−0.791 + 0.611i)16-s + (−0.520 − 0.853i)17-s + ⋯
L(s)  = 1  + (−0.581 + 0.813i)2-s + (−0.976 − 0.217i)3-s + (−0.322 − 0.946i)4-s + (0.639 − 0.768i)5-s + (0.744 − 0.667i)6-s + (−0.905 − 0.424i)7-s + (0.957 + 0.288i)8-s + (0.905 + 0.424i)9-s + (0.252 + 0.967i)10-s + (0.520 + 0.853i)11-s + (0.109 + 0.994i)12-s + (0.581 + 0.813i)13-s + (0.872 − 0.489i)14-s + (−0.791 + 0.611i)15-s + (−0.791 + 0.611i)16-s + (−0.520 − 0.853i)17-s + ⋯

Functional equation

Λ(s)=(431s/2ΓR(s+1)L(s)=((0.114+0.993i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 431 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.114 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(431s/2ΓR(s+1)L(s)=((0.114+0.993i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 431 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.114 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 11
Conductor: 431431
Sign: 0.114+0.993i-0.114 + 0.993i
Analytic conductor: 46.317346.3173
Root analytic conductor: 46.317346.3173
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ431(367,)\chi_{431} (367, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (1, 431, (1: ), 0.114+0.993i)(1,\ 431,\ (1:\ ),\ -0.114 + 0.993i)

Particular Values

L(12)L(\frac{1}{2}) \approx 0.4531352824+0.5084804492i0.4531352824 + 0.5084804492i
L(12)L(\frac12) \approx 0.4531352824+0.5084804492i0.4531352824 + 0.5084804492i
L(1)L(1) \approx 0.5796830088+0.1267796929i0.5796830088 + 0.1267796929i
L(1)L(1) \approx 0.5796830088+0.1267796929i0.5796830088 + 0.1267796929i

Euler product

   L(s)=pFp(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad431 1 1
good2 1+(0.581+0.813i)T 1 + (-0.581 + 0.813i)T
3 1+(0.9760.217i)T 1 + (-0.976 - 0.217i)T
5 1+(0.6390.768i)T 1 + (0.639 - 0.768i)T
7 1+(0.9050.424i)T 1 + (-0.905 - 0.424i)T
11 1+(0.520+0.853i)T 1 + (0.520 + 0.853i)T
13 1+(0.581+0.813i)T 1 + (0.581 + 0.813i)T
17 1+(0.5200.853i)T 1 + (-0.520 - 0.853i)T
19 1+(0.639+0.768i)T 1 + (0.639 + 0.768i)T
23 1+(0.03650.999i)T 1 + (-0.0365 - 0.999i)T
29 1+(0.322+0.946i)T 1 + (-0.322 + 0.946i)T
31 1+(0.7440.667i)T 1 + (-0.744 - 0.667i)T
37 1+(0.8720.489i)T 1 + (0.872 - 0.489i)T
41 1+(0.791+0.611i)T 1 + (-0.791 + 0.611i)T
43 1+(0.252+0.967i)T 1 + (-0.252 + 0.967i)T
47 1+(0.989+0.145i)T 1 + (-0.989 + 0.145i)T
53 1+(0.989+0.145i)T 1 + (0.989 + 0.145i)T
59 1+(0.581+0.813i)T 1 + (-0.581 + 0.813i)T
61 1+(0.6940.719i)T 1 + (-0.694 - 0.719i)T
67 1+(0.8720.489i)T 1 + (0.872 - 0.489i)T
71 1+(0.989+0.145i)T 1 + (-0.989 + 0.145i)T
73 1+(0.3910.920i)T 1 + (-0.391 - 0.920i)T
79 1+(0.833+0.551i)T 1 + (-0.833 + 0.551i)T
83 1+(0.976+0.217i)T 1 + (0.976 + 0.217i)T
89 1+(0.934+0.357i)T 1 + (0.934 + 0.357i)T
97 1+(0.833+0.551i)T 1 + (0.833 + 0.551i)T
show more
show less
   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

−23.3219859774471935958058063443, −22.444486646877790610444131691012, −21.87641327483486464704154181685, −21.50652357483713015035318990433, −20.16007182230975209542186314915, −19.18081085287920758966878416644, −18.48000717244407418499972206988, −17.70573826557487865038619681566, −17.04608342985136205716872738287, −16.04395028241143748280506970767, −15.20921810132060322001630035248, −13.49665600180046084382113792881, −13.08499426974492885210740640572, −11.83122375514205885786744758040, −11.151153975119400456303733691232, −10.35813748852968084070271617467, −9.61813151664264416437334221562, −8.72266240450363159276766920189, −7.20832261525262466745142597609, −6.266404484287868325411459578241, −5.46680958431954477206429614341, −3.771426158136544956701515688046, −3.07798442829716368398996184886, −1.641436984638840286186835872540, −0.33450073360257433565298754872, 0.89879251394723425784393922605, 1.83596406010170896389605595472, 4.20809433026769274678871054423, 5.02335106616191417405280780990, 6.1595546855002009544894627062, 6.65966683070242713274637630487, 7.629406883218984912095104814778, 9.120588293400596742799264135307, 9.62874465801180879165644085793, 10.52929702432839516911906102759, 11.71966852318756127447844878332, 12.83597781529211140018259746536, 13.52480922289832581182265317325, 14.57467869010000778832689979411, 16.00332772576868039653040347519, 16.49860175195297351336064908283, 16.926686913782899760482604738962, 18.11758535312219673699477645946, 18.43817133550442010821400032521, 19.79339577580632745209913225705, 20.484908254777856498600620158953, 21.89430597944708043069211313198, 22.75035439374288407205342680275, 23.29766607757294342778765622896, 24.32088034503486797845544290654

Graph of the ZZ-function along the critical line