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Properties

Label 1-431-431.144-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: $\chi_{431} (144, \cdot )$
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

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

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