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Properties

Label 1-431-431.367-r1-0-0
Degree $1$
Conductor $431$
Sign $-0.114 + 0.993i$
Analytic cond. $46.3173$
Root an. cond. $46.3173$
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  + (−0.581 + 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

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

Invariants

Degree: \(1\)
Conductor: \(431\)
Sign: $-0.114 + 0.993i$
Analytic conductor: \(46.3173\)
Root analytic conductor: \(46.3173\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{431} (367, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 431,\ (1:\ ),\ -0.114 + 0.993i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4531352824 + 0.5084804492i\)
\(L(\frac12)\) \(\approx\) \(0.4531352824 + 0.5084804492i\)
\(L(1)\) \(\approx\) \(0.5796830088 + 0.1267796929i\)
\(L(1)\) \(\approx\) \(0.5796830088 + 0.1267796929i\)

Euler product

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

−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 $Z$-function along the critical line