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Properties

Label 1-431-431.163-r0-0-0
Degree $1$
Conductor $431$
Sign $0.792 + 0.610i$
Analytic cond. $2.00155$
Root an. cond. $2.00155$
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  + (0.391 + 0.920i)2-s + (−0.872 + 0.489i)3-s + (−0.694 + 0.719i)4-s + (−0.987 − 0.160i)5-s + (−0.791 − 0.611i)6-s + (−0.923 + 0.384i)7-s + (−0.934 − 0.357i)8-s + (0.520 − 0.853i)9-s + (−0.238 − 0.971i)10-s + (0.0219 − 0.999i)11-s + (0.252 − 0.967i)12-s + (0.996 + 0.0875i)13-s + (−0.714 − 0.699i)14-s + (0.939 − 0.343i)15-s + (−0.0365 − 0.999i)16-s + (−0.944 − 0.329i)17-s + ⋯
L(s)  = 1  + (0.391 + 0.920i)2-s + (−0.872 + 0.489i)3-s + (−0.694 + 0.719i)4-s + (−0.987 − 0.160i)5-s + (−0.791 − 0.611i)6-s + (−0.923 + 0.384i)7-s + (−0.934 − 0.357i)8-s + (0.520 − 0.853i)9-s + (−0.238 − 0.971i)10-s + (0.0219 − 0.999i)11-s + (0.252 − 0.967i)12-s + (0.996 + 0.0875i)13-s + (−0.714 − 0.699i)14-s + (0.939 − 0.343i)15-s + (−0.0365 − 0.999i)16-s + (−0.944 − 0.329i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 431 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.792 + 0.610i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 431 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.792 + 0.610i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(431\)
Sign: $0.792 + 0.610i$
Analytic conductor: \(2.00155\)
Root analytic conductor: \(2.00155\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{431} (163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 431,\ (0:\ ),\ 0.792 + 0.610i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5601834679 + 0.1906648099i\)
\(L(\frac12)\) \(\approx\) \(0.5601834679 + 0.1906648099i\)
\(L(1)\) \(\approx\) \(0.5729981281 + 0.3295645389i\)
\(L(1)\) \(\approx\) \(0.5729981281 + 0.3295645389i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad431 \( 1 \)
good2 \( 1 + (0.391 + 0.920i)T \)
3 \( 1 + (-0.872 + 0.489i)T \)
5 \( 1 + (-0.987 - 0.160i)T \)
7 \( 1 + (-0.923 + 0.384i)T \)
11 \( 1 + (0.0219 - 0.999i)T \)
13 \( 1 + (0.996 + 0.0875i)T \)
17 \( 1 + (-0.944 - 0.329i)T \)
19 \( 1 + (0.892 + 0.450i)T \)
23 \( 1 + (-0.982 - 0.188i)T \)
29 \( 1 + (-0.899 + 0.437i)T \)
31 \( 1 + (0.281 - 0.959i)T \)
37 \( 1 + (0.167 + 0.985i)T \)
41 \( 1 + (0.616 + 0.787i)T \)
43 \( 1 + (-0.238 + 0.971i)T \)
47 \( 1 + (-0.181 - 0.983i)T \)
53 \( 1 + (0.724 - 0.688i)T \)
59 \( 1 + (-0.857 - 0.514i)T \)
61 \( 1 + (0.917 - 0.397i)T \)
67 \( 1 + (0.444 - 0.895i)T \)
71 \( 1 + (0.879 - 0.476i)T \)
73 \( 1 + (0.495 - 0.868i)T \)
79 \( 1 + (-0.404 - 0.914i)T \)
83 \( 1 + (0.195 + 0.980i)T \)
89 \( 1 + (0.817 + 0.575i)T \)
97 \( 1 + (0.864 - 0.502i)T \)
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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

−23.71518905964672375508358242299, −22.81712412167486005562657203293, −22.72047647329243641821724788580, −21.71796609658216025359064670814, −20.29771717240603515641872240102, −19.84510504414242713836526736942, −18.96729129041815059511753739628, −18.172236975163206616996794855310, −17.3973548286784024344777969069, −15.95392162382026230677253459383, −15.54454174827342094579600290382, −14.055240643879170084311556881001, −13.12286415800679906324018791162, −12.49557891414127810966006170308, −11.67325772359077184722033661301, −10.89800373064375728895498199790, −10.13815838576808431499927331881, −8.95251586272076591708822591422, −7.53542671948747208925486353071, −6.64962758722066470699301221473, −5.62220612562976152944233371495, −4.353538278515262380001446550662, −3.70292186739846732624759514101, −2.28331601288932859821300177784, −0.888362641869654542286265821061, 0.47554487890210284620083406752, 3.318713729549380846153051935150, 3.870432461700045318118952127300, 5.00049960760000805644123492141, 6.049355775790354952914371639391, 6.57582238661155276293743612537, 7.87225613830724453547880029433, 8.831374196288545854153853339660, 9.72627574812254057786706982246, 11.21210531147554760263583629507, 11.79714790654854949835455047815, 12.82305851403637588216968549004, 13.624675623568424991453149231701, 14.99589471772240276735845796271, 15.76894591291070575189996032774, 16.25277494947913073261036085688, 16.74145201324455203431131875798, 18.25091759012019493974927644939, 18.58745641827970290195114225461, 20.0023792130638015719357873130, 21.0741928453833857687176635234, 22.159760014560664602165001316822, 22.516588197509461986630317093653, 23.33579257724673881707543484965, 24.115188914852296968029818198270

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