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Properties

Label 1-1840-1840.1699-r0-0-0
Degree $1$
Conductor $1840$
Sign $0.638 - 0.769i$
Analytic cond. $8.54492$
Root an. cond. $8.54492$
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  + (0.755 − 0.654i)3-s + (0.415 − 0.909i)7-s + (0.142 − 0.989i)9-s + (−0.281 + 0.959i)11-s + (0.909 − 0.415i)13-s + (0.841 + 0.540i)17-s + (0.540 + 0.841i)19-s + (−0.281 − 0.959i)21-s + (−0.540 − 0.841i)27-s + (−0.540 + 0.841i)29-s + (0.654 − 0.755i)31-s + (0.415 + 0.909i)33-s + (0.989 + 0.142i)37-s + (0.415 − 0.909i)39-s + (0.142 + 0.989i)41-s + ⋯
L(s)  = 1  + (0.755 − 0.654i)3-s + (0.415 − 0.909i)7-s + (0.142 − 0.989i)9-s + (−0.281 + 0.959i)11-s + (0.909 − 0.415i)13-s + (0.841 + 0.540i)17-s + (0.540 + 0.841i)19-s + (−0.281 − 0.959i)21-s + (−0.540 − 0.841i)27-s + (−0.540 + 0.841i)29-s + (0.654 − 0.755i)31-s + (0.415 + 0.909i)33-s + (0.989 + 0.142i)37-s + (0.415 − 0.909i)39-s + (0.142 + 0.989i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.638 - 0.769i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.638 - 0.769i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1840\)    =    \(2^{4} \cdot 5 \cdot 23\)
Sign: $0.638 - 0.769i$
Analytic conductor: \(8.54492\)
Root analytic conductor: \(8.54492\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1840} (1699, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1840,\ (0:\ ),\ 0.638 - 0.769i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.283909108 - 1.073535611i\)
\(L(\frac12)\) \(\approx\) \(2.283909108 - 1.073535611i\)
\(L(1)\) \(\approx\) \(1.500562670 - 0.4240162046i\)
\(L(1)\) \(\approx\) \(1.500562670 - 0.4240162046i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
23 \( 1 \)
good3 \( 1 + (0.755 - 0.654i)T \)
7 \( 1 + (0.415 - 0.909i)T \)
11 \( 1 + (-0.281 + 0.959i)T \)
13 \( 1 + (0.909 - 0.415i)T \)
17 \( 1 + (0.841 + 0.540i)T \)
19 \( 1 + (0.540 + 0.841i)T \)
29 \( 1 + (-0.540 + 0.841i)T \)
31 \( 1 + (0.654 - 0.755i)T \)
37 \( 1 + (0.989 + 0.142i)T \)
41 \( 1 + (0.142 + 0.989i)T \)
43 \( 1 + (0.755 - 0.654i)T \)
47 \( 1 + T \)
53 \( 1 + (-0.909 - 0.415i)T \)
59 \( 1 + (-0.909 + 0.415i)T \)
61 \( 1 + (0.755 + 0.654i)T \)
67 \( 1 + (0.281 + 0.959i)T \)
71 \( 1 + (-0.959 + 0.281i)T \)
73 \( 1 + (0.841 - 0.540i)T \)
79 \( 1 + (0.415 + 0.909i)T \)
83 \( 1 + (0.989 + 0.142i)T \)
89 \( 1 + (-0.654 - 0.755i)T \)
97 \( 1 + (-0.142 - 0.989i)T \)
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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

−20.511144784784744056594080369707, −19.396142185380147346769912209266, −18.85219013048617611123961485759, −18.292289464053841465376757391195, −17.26242316381643674298501103909, −16.24729725206546735746522953356, −15.823893633023787238914211692140, −15.21645665106053345266654571499, −14.18256564082103273302554834171, −13.87646818308198471769186821937, −12.9904570222982540425713641025, −11.910549669246630276290165465096, −11.18967654304418751753980290495, −10.591219491765062720136857086459, −9.30146035515083995642381933487, −9.1759192662873775515125712072, −8.143427087815929388043858031547, −7.68789263951643438445696502908, −6.32785427948918288868129704046, −5.508927675656311157749457217625, −4.81062863636767853055200824563, −3.78600580854952299865104998746, −2.98280033215199421194454620381, −2.313309443805735365652191399977, −1.083657658971069382305304960399, 1.03578406111463626854869709713, 1.590526799601587942040831492954, 2.72245361932926218150858994850, 3.66627812399939749459432194835, 4.27438292578319849808485692729, 5.51671116892460595370075299017, 6.367433478778091543934455524415, 7.3886678415078155979932228932, 7.77769828594537471812298724753, 8.460956054085695322198758162551, 9.56186999789970494397639005551, 10.16901109121690073765256088843, 11.03716840832344123626089127606, 12.02061467011533994785340366652, 12.77118037844001198528513579943, 13.352021957007977509368500340527, 14.15684419383069905435915994913, 14.71551590636647428297054067603, 15.43780895455469980560889163052, 16.4329263516648800698666656694, 17.23224034837543605752331573002, 18.01763033185300198103986298927, 18.50339037514320006389587302259, 19.336957169788946818111749755081, 20.22167726447243162875707805823

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