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Properties

Label 1-1840-1840.493-r0-0-0
Degree $1$
Conductor $1840$
Sign $-0.837 - 0.546i$
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  + (−0.415 − 0.909i)3-s + (0.540 − 0.841i)7-s + (−0.654 + 0.755i)9-s + (0.989 − 0.142i)11-s + (−0.841 + 0.540i)13-s + (−0.281 − 0.959i)17-s + (0.281 − 0.959i)19-s + (−0.989 − 0.142i)21-s + (0.959 + 0.281i)27-s + (0.281 + 0.959i)29-s + (0.415 − 0.909i)31-s + (−0.540 − 0.841i)33-s + (−0.654 + 0.755i)37-s + (0.841 + 0.540i)39-s + (0.654 + 0.755i)41-s + ⋯
L(s)  = 1  + (−0.415 − 0.909i)3-s + (0.540 − 0.841i)7-s + (−0.654 + 0.755i)9-s + (0.989 − 0.142i)11-s + (−0.841 + 0.540i)13-s + (−0.281 − 0.959i)17-s + (0.281 − 0.959i)19-s + (−0.989 − 0.142i)21-s + (0.959 + 0.281i)27-s + (0.281 + 0.959i)29-s + (0.415 − 0.909i)31-s + (−0.540 − 0.841i)33-s + (−0.654 + 0.755i)37-s + (0.841 + 0.540i)39-s + (0.654 + 0.755i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.837 - 0.546i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.837 - 0.546i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1840\)    =    \(2^{4} \cdot 5 \cdot 23\)
Sign: $-0.837 - 0.546i$
Analytic conductor: \(8.54492\)
Root analytic conductor: \(8.54492\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1840} (493, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1840,\ (0:\ ),\ -0.837 - 0.546i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3293600847 - 1.107498526i\)
\(L(\frac12)\) \(\approx\) \(0.3293600847 - 1.107498526i\)
\(L(1)\) \(\approx\) \(0.8217101909 - 0.4568847778i\)
\(L(1)\) \(\approx\) \(0.8217101909 - 0.4568847778i\)

Euler product

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

−20.64513144802136842750727872406, −19.67569020240131622521888751266, −19.14324568295213280332815413394, −17.94176150681847989343456023050, −17.45467410075668987029942793876, −16.905999573542121861149162240315, −15.87687904451237919175840483529, −15.38560915969800132437875384794, −14.463869603980807149339211817948, −14.31105040031012598741883021236, −12.69606627282873511263967887703, −12.16131546049046804501042035033, −11.53145184410601376991228094386, −10.673680097963963259397937138468, −9.95086932841551116937506020675, −9.2121673856056472636980123602, −8.500113475460885744435272266998, −7.657484710877904323439161283512, −6.38389465528062568950768763934, −5.83188073947490648712535709545, −4.97924189499100760586123262174, −4.254938986748082985100283017244, −3.39474650781050329142563738860, −2.36215161034035247355413575285, −1.265234600831175604682303932321, 0.46189035053792670092969173314, 1.3921033894397406299056609214, 2.26917742235202794259568955040, 3.33745961961022810521071093162, 4.60472262587866717209053620252, 5.01259170230876741691361485489, 6.268433074457851567652573194351, 7.01379215857891784704407756246, 7.3516427624784277939380151808, 8.39591028713824477073335659094, 9.21786647395117069604596727604, 10.14129218917002339857629232820, 11.20141030621896211412755513461, 11.57842689849133402916573052119, 12.272930987159115805940649427999, 13.29120458420062779759953389171, 13.90433144660150114356147099615, 14.371447135603686748022173472791, 15.38335374589818144363007833420, 16.60935588269164271084786289851, 16.88968124059910659889849731867, 17.66892362820490340989015547315, 18.25296770427981498566450848613, 19.162487871923746979533836278938, 19.84161259042157961444226195381

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