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Properties

Label 1-431-431.3-r0-0-0
Degree $1$
Conductor $431$
Sign $-0.751 + 0.659i$
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  + (−0.0365 + 0.999i)2-s + (−0.457 + 0.889i)3-s + (−0.997 − 0.0729i)4-s + (−0.322 − 0.946i)5-s + (−0.872 − 0.489i)6-s + (−0.581 − 0.813i)7-s + (0.109 − 0.994i)8-s + (−0.581 − 0.813i)9-s + (0.957 − 0.288i)10-s + (0.391 + 0.920i)11-s + (0.520 − 0.853i)12-s + (−0.0365 − 0.999i)13-s + (0.833 − 0.551i)14-s + (0.989 + 0.145i)15-s + (0.989 + 0.145i)16-s + (0.391 + 0.920i)17-s + ⋯
L(s)  = 1  + (−0.0365 + 0.999i)2-s + (−0.457 + 0.889i)3-s + (−0.997 − 0.0729i)4-s + (−0.322 − 0.946i)5-s + (−0.872 − 0.489i)6-s + (−0.581 − 0.813i)7-s + (0.109 − 0.994i)8-s + (−0.581 − 0.813i)9-s + (0.957 − 0.288i)10-s + (0.391 + 0.920i)11-s + (0.520 − 0.853i)12-s + (−0.0365 − 0.999i)13-s + (0.833 − 0.551i)14-s + (0.989 + 0.145i)15-s + (0.989 + 0.145i)16-s + (0.391 + 0.920i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 431 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.751 + 0.659i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 431 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.751 + 0.659i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(431\)
Sign: $-0.751 + 0.659i$
Analytic conductor: \(2.00155\)
Root analytic conductor: \(2.00155\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{431} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 431,\ (0:\ ),\ -0.751 + 0.659i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2280148289 + 0.6058136427i\)
\(L(\frac12)\) \(\approx\) \(0.2280148289 + 0.6058136427i\)
\(L(1)\) \(\approx\) \(0.5508660431 + 0.4034096339i\)
\(L(1)\) \(\approx\) \(0.5508660431 + 0.4034096339i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad431 \( 1 \)
good2 \( 1 + (-0.0365 + 0.999i)T \)
3 \( 1 + (-0.457 + 0.889i)T \)
5 \( 1 + (-0.322 - 0.946i)T \)
7 \( 1 + (-0.581 - 0.813i)T \)
11 \( 1 + (0.391 + 0.920i)T \)
13 \( 1 + (-0.0365 - 0.999i)T \)
17 \( 1 + (0.391 + 0.920i)T \)
19 \( 1 + (-0.322 + 0.946i)T \)
23 \( 1 + (-0.181 + 0.983i)T \)
29 \( 1 + (-0.997 + 0.0729i)T \)
31 \( 1 + (-0.872 + 0.489i)T \)
37 \( 1 + (0.833 - 0.551i)T \)
41 \( 1 + (0.989 + 0.145i)T \)
43 \( 1 + (0.957 + 0.288i)T \)
47 \( 1 + (0.744 + 0.667i)T \)
53 \( 1 + (0.744 - 0.667i)T \)
59 \( 1 + (-0.0365 + 0.999i)T \)
61 \( 1 + (0.639 - 0.768i)T \)
67 \( 1 + (0.833 - 0.551i)T \)
71 \( 1 + (0.744 + 0.667i)T \)
73 \( 1 + (0.905 + 0.424i)T \)
79 \( 1 + (-0.976 + 0.217i)T \)
83 \( 1 + (-0.457 + 0.889i)T \)
89 \( 1 + (0.252 + 0.967i)T \)
97 \( 1 + (-0.976 - 0.217i)T \)
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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

−23.63995260540208003355225423482, −22.66512878868627414026567149665, −22.15933402974447589250468064409, −21.51539635466249989371911851763, −20.082944778340854545047251834319, −19.20420091898953814979228037664, −18.67661117868935772745832549240, −18.33439159510823620372184589259, −17.06260985539884228958581453657, −16.16449776344541367629364044366, −14.680881558626799723849079472953, −13.93161325338260789295474606815, −13.051989876770988364736152426368, −12.06154603721452078078110961172, −11.43813041495739150330207990381, −10.8503598461219170935275314684, −9.4784308868153444904679868496, −8.66524513977027636019466069297, −7.420290477606564628909916084590, −6.41623023144290413720589754520, −5.51926042541671751615580356825, −4.04533224833505043701868155170, −2.804142147657862863978442239843, −2.20792747485161425009425395729, −0.50192577717724597942589963817, 1.04745201174470673970714698904, 3.817718608763178993999573803874, 4.03151391959290391651483202368, 5.33511639256526256840313939094, 5.9569290302633217567301559845, 7.30473058509619673151550758866, 8.13676528347885176745331373034, 9.36266879121530453754068908600, 9.85073577393730625502625330476, 10.89682808796338087844536101203, 12.52949128787648012236836564716, 12.80720196194067350390805198143, 14.27192069947629645037058321536, 15.10057557741596552234263548696, 15.86812335772488384712463721343, 16.661307523851270495288290408901, 17.15186645354146573858384739343, 17.91041399358973301842055641181, 19.46222786126031161456757970214, 20.15257849234286488383918476128, 21.104991495831222092093132555998, 22.17529092906458212963160954878, 23.10076726377564953593299101203, 23.31972353079280616332354603385, 24.407744636262877858578657878119

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