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Properties

Label 1-1840-1840.1227-r0-0-0
Degree $1$
Conductor $1840$
Sign $-0.869 + 0.493i$
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  + (−0.654 + 0.755i)3-s + (−0.909 + 0.415i)7-s + (−0.142 − 0.989i)9-s + (0.281 + 0.959i)11-s + (−0.415 + 0.909i)13-s + (0.540 + 0.841i)17-s + (0.540 − 0.841i)19-s + (0.281 − 0.959i)21-s + (0.841 + 0.540i)27-s + (0.540 + 0.841i)29-s + (0.654 + 0.755i)31-s + (−0.909 − 0.415i)33-s + (0.142 + 0.989i)37-s + (−0.415 − 0.909i)39-s + (0.142 − 0.989i)41-s + ⋯
L(s)  = 1  + (−0.654 + 0.755i)3-s + (−0.909 + 0.415i)7-s + (−0.142 − 0.989i)9-s + (0.281 + 0.959i)11-s + (−0.415 + 0.909i)13-s + (0.540 + 0.841i)17-s + (0.540 − 0.841i)19-s + (0.281 − 0.959i)21-s + (0.841 + 0.540i)27-s + (0.540 + 0.841i)29-s + (0.654 + 0.755i)31-s + (−0.909 − 0.415i)33-s + (0.142 + 0.989i)37-s + (−0.415 − 0.909i)39-s + (0.142 − 0.989i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.869 + 0.493i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.869 + 0.493i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1840\)    =    \(2^{4} \cdot 5 \cdot 23\)
Sign: $-0.869 + 0.493i$
Analytic conductor: \(8.54492\)
Root analytic conductor: \(8.54492\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1840} (1227, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1840,\ (0:\ ),\ -0.869 + 0.493i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2340847796 + 0.8864612646i\)
\(L(\frac12)\) \(\approx\) \(0.2340847796 + 0.8864612646i\)
\(L(1)\) \(\approx\) \(0.6943617827 + 0.3740687859i\)
\(L(1)\) \(\approx\) \(0.6943617827 + 0.3740687859i\)

Euler product

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

−19.53148116254147919245190721469, −19.19484329930901549445138431133, −18.37120201550187018782934364186, −17.66835006123175779122604009325, −16.853179453774449972331237720233, −16.31420319410416243065264196080, −15.71710388732716916777231352995, −14.412009200575650639388801125357, −13.81037443951763427841133931736, −13.07800990876327112584986741701, −12.44688901655714769375164189956, −11.68287648051528205297395810391, −10.98021068609801078369519263022, −10.05113151871378746859276370898, −9.50173916285419383962044992912, −8.1301086734664915555419031375, −7.69731591012010266940368485714, −6.7646122328096143022642351719, −5.991507122584851622885051858822, −5.501302469435631263643265393056, −4.34697091863107856514409498118, −3.22781473233150632515156167079, −2.57486547555078115865152114021, −1.11861431445095151029160536943, −0.44859084701094896658143768340, 1.14467124484306481827959132385, 2.43324035395806841025850723457, 3.396536040728746803534577415931, 4.23907548953694933903202160999, 4.997250164100016750566553821230, 5.82205870102010025206922308904, 6.71868307906411533333081177045, 7.18071998010812054485734656470, 8.73221247741433337276169065946, 9.20308397370991645561182139324, 10.08174962913239060989833673887, 10.45054158541196654863764454113, 11.7350894854924779715110423786, 12.082212232554726473560619436040, 12.79763435429706425716941748561, 13.88258715087388680360211302845, 14.75938342149434538095965034961, 15.413601766158990632422750870475, 16.02509468037188754868854831570, 16.78804436069615072355837232815, 17.364014375551985435233679782258, 18.129880318805184279363158232395, 19.04933998375034871445533120067, 19.729705834996993384981294591307, 20.473238885377434113637834446903

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