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Properties

Label 1-431-431.87-r0-0-0
Degree $1$
Conductor $431$
Sign $0.551 + 0.834i$
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  + (0.639 + 0.768i)2-s + (0.391 − 0.920i)3-s + (−0.181 + 0.983i)4-s + (0.939 − 0.343i)5-s + (0.957 − 0.288i)6-s + (0.138 + 0.990i)7-s + (−0.872 + 0.489i)8-s + (−0.694 − 0.719i)9-s + (0.864 + 0.502i)10-s + (−0.0948 + 0.995i)11-s + (0.833 + 0.551i)12-s + (0.928 + 0.370i)13-s + (−0.672 + 0.739i)14-s + (0.0511 − 0.998i)15-s + (−0.934 − 0.357i)16-s + (0.917 + 0.397i)17-s + ⋯
L(s)  = 1  + (0.639 + 0.768i)2-s + (0.391 − 0.920i)3-s + (−0.181 + 0.983i)4-s + (0.939 − 0.343i)5-s + (0.957 − 0.288i)6-s + (0.138 + 0.990i)7-s + (−0.872 + 0.489i)8-s + (−0.694 − 0.719i)9-s + (0.864 + 0.502i)10-s + (−0.0948 + 0.995i)11-s + (0.833 + 0.551i)12-s + (0.928 + 0.370i)13-s + (−0.672 + 0.739i)14-s + (0.0511 − 0.998i)15-s + (−0.934 − 0.357i)16-s + (0.917 + 0.397i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 431 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.551 + 0.834i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 431 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.551 + 0.834i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(431\)
Sign: $0.551 + 0.834i$
Analytic conductor: \(2.00155\)
Root analytic conductor: \(2.00155\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{431} (87, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 431,\ (0:\ ),\ 0.551 + 0.834i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.095045530 + 1.126907094i\)
\(L(\frac12)\) \(\approx\) \(2.095045530 + 1.126907094i\)
\(L(1)\) \(\approx\) \(1.719898188 + 0.5629544708i\)
\(L(1)\) \(\approx\) \(1.719898188 + 0.5629544708i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad431 \( 1 \)
good2 \( 1 + (0.639 + 0.768i)T \)
3 \( 1 + (0.391 - 0.920i)T \)
5 \( 1 + (0.939 - 0.343i)T \)
7 \( 1 + (0.138 + 0.990i)T \)
11 \( 1 + (-0.0948 + 0.995i)T \)
13 \( 1 + (0.928 + 0.370i)T \)
17 \( 1 + (0.917 + 0.397i)T \)
19 \( 1 + (-0.557 - 0.829i)T \)
23 \( 1 + (-0.295 + 0.955i)T \)
29 \( 1 + (-0.991 - 0.131i)T \)
31 \( 1 + (-0.944 + 0.329i)T \)
37 \( 1 + (0.979 - 0.203i)T \)
41 \( 1 + (0.965 - 0.259i)T \)
43 \( 1 + (0.864 - 0.502i)T \)
47 \( 1 + (0.252 - 0.967i)T \)
53 \( 1 + (0.364 - 0.931i)T \)
59 \( 1 + (-0.969 - 0.245i)T \)
61 \( 1 + (0.948 + 0.315i)T \)
67 \( 1 + (-0.911 - 0.411i)T \)
71 \( 1 + (0.998 - 0.0584i)T \)
73 \( 1 + (-0.152 + 0.988i)T \)
79 \( 1 + (0.972 - 0.231i)T \)
83 \( 1 + (-0.754 - 0.656i)T \)
89 \( 1 + (-0.885 + 0.463i)T \)
97 \( 1 + (-0.650 - 0.759i)T \)
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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

−23.747025123049228242704215376150, −22.80852766513078995080318127502, −22.212333897298720273319450240912, −21.123663933587307342774333767929, −20.89504640880228134362467412566, −20.09075924345373156159018714360, −18.94152298675279896773763648199, −18.21448206101056318886541892031, −16.81307635585754784770277681223, −16.20244434377558849621103015459, −14.79721406547588369299462465695, −14.2465231142667916307892775383, −13.621972620176842171408449492752, −12.760172235981727317154076604149, −11.01851233086937339212007255097, −10.838871368456825581368487444286, −9.92876033959524619288793132073, −9.11096332798836422583282732715, −7.8665538985532140414468957587, −6.12980833353968773330108174049, −5.5816564951025207627429752141, −4.27396891246431251171513666478, −3.49472179282439352487538776044, −2.59016257847065798177452807492, −1.18924534613989755464314158214, 1.72177831316162231624368164117, 2.52580186903815282361080272430, 3.88654512139157223093026213195, 5.373377388500501432802731767796, 5.90572082024981742524998653132, 6.88329551802130808359961625082, 7.87299460218519715630571771328, 8.8678741543529315929157627919, 9.43884888500696794089259964494, 11.31740292232811889450732788910, 12.41164970240620185538355705234, 12.88256558891878340825080575674, 13.702057472307158294264634103105, 14.60231091981498786845161881840, 15.25783120679770752812127045813, 16.43668993041892353110909172074, 17.48853018757880003399533603592, 17.99260592940545515769173597189, 18.8067890288455862967089549909, 20.15549412928229488683548236079, 21.08421579911755680071304974525, 21.633365960218765092405940114273, 22.75340367541647406477222871511, 23.67411121924880586172385233373, 24.244038167943755454492496041191

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