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Properties

Label 1-1840-1840.1269-r0-0-0
Degree $1$
Conductor $1840$
Sign $0.846 - 0.531i$
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  + (0.540 + 0.841i)3-s + (−0.959 + 0.281i)7-s + (−0.415 + 0.909i)9-s + (0.755 − 0.654i)11-s + (0.281 − 0.959i)13-s + (0.142 − 0.989i)17-s + (−0.989 + 0.142i)19-s + (−0.755 − 0.654i)21-s + (−0.989 + 0.142i)27-s + (0.989 + 0.142i)29-s + (0.841 + 0.540i)31-s + (0.959 + 0.281i)33-s + (−0.909 − 0.415i)37-s + (0.959 − 0.281i)39-s + (−0.415 − 0.909i)41-s + ⋯
L(s)  = 1  + (0.540 + 0.841i)3-s + (−0.959 + 0.281i)7-s + (−0.415 + 0.909i)9-s + (0.755 − 0.654i)11-s + (0.281 − 0.959i)13-s + (0.142 − 0.989i)17-s + (−0.989 + 0.142i)19-s + (−0.755 − 0.654i)21-s + (−0.989 + 0.142i)27-s + (0.989 + 0.142i)29-s + (0.841 + 0.540i)31-s + (0.959 + 0.281i)33-s + (−0.909 − 0.415i)37-s + (0.959 − 0.281i)39-s + (−0.415 − 0.909i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.846 - 0.531i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.846 - 0.531i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1840\)    =    \(2^{4} \cdot 5 \cdot 23\)
Sign: $0.846 - 0.531i$
Analytic conductor: \(8.54492\)
Root analytic conductor: \(8.54492\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1840} (1269, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1840,\ (0:\ ),\ 0.846 - 0.531i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.356860146 - 0.3905809969i\)
\(L(\frac12)\) \(\approx\) \(1.356860146 - 0.3905809969i\)
\(L(1)\) \(\approx\) \(1.097877497 + 0.1188524202i\)
\(L(1)\) \(\approx\) \(1.097877497 + 0.1188524202i\)

Euler product

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

−19.88613370481723163984519150857, −19.418050664241051494485401114269, −19.04425499415825948360410403597, −18.07239823247211273324963049158, −17.245063843894311002508653507018, −16.74341377578318121243206363880, −15.7267978884076957052406184584, −14.84808824292322444901353359101, −14.333550698567747862020575479718, −13.33568777947528582852762649894, −12.99242132671812080954520866189, −12.07646805974594522519163320717, −11.54040173850208053990431874108, −10.23356381552945136330524865881, −9.67481221197178494213832731967, −8.71991380735594124518879143791, −8.22702069993092546721719899497, −6.9880902326865899196254615544, −6.62262133793230031953982936683, −6.03111320216093958149289059529, −4.46382063044438224245660008140, −3.82258605273606100885939475129, −2.88576441939912164165514441237, −1.92615883458385977989212454114, −1.128059709593818058577243159315, 0.4891455011170566371202354542, 2.04385395249020140365800997403, 3.179781889053371084663136109268, 3.38481763873032768121769709028, 4.52220183384385125807663269555, 5.39704626183182493751008503500, 6.233212720470521596300166905486, 7.06096617801118498755430404102, 8.31141180000920732875636694663, 8.71650375520440875128439723681, 9.547051949120292117565572129732, 10.26088764976405750349186051154, 10.87228496283236118377786606104, 11.91787716139595677776664049433, 12.644886323273712631465616835257, 13.70080894675883352652794754426, 14.02591599609579864084231856109, 15.12614925642009407728044161641, 15.62456005575805446457531270579, 16.29074743634739904120542280242, 16.933250803223652070900072981632, 17.830642433847532478578017967412, 18.93517378686514948283340009369, 19.36801731608538768965900312407, 20.063957411331905133369531610426

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