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Properties

Label 1-1840-1840.1067-r0-0-0
Degree $1$
Conductor $1840$
Sign $0.536 - 0.843i$
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  + (−0.142 + 0.989i)3-s + (0.755 − 0.654i)7-s + (−0.959 − 0.281i)9-s + (0.540 − 0.841i)11-s + (0.654 − 0.755i)13-s + (−0.909 + 0.415i)17-s + (−0.909 − 0.415i)19-s + (0.540 + 0.841i)21-s + (0.415 − 0.909i)27-s + (−0.909 + 0.415i)29-s + (0.142 + 0.989i)31-s + (0.755 + 0.654i)33-s + (0.959 + 0.281i)37-s + (0.654 + 0.755i)39-s + (0.959 − 0.281i)41-s + ⋯
L(s)  = 1  + (−0.142 + 0.989i)3-s + (0.755 − 0.654i)7-s + (−0.959 − 0.281i)9-s + (0.540 − 0.841i)11-s + (0.654 − 0.755i)13-s + (−0.909 + 0.415i)17-s + (−0.909 − 0.415i)19-s + (0.540 + 0.841i)21-s + (0.415 − 0.909i)27-s + (−0.909 + 0.415i)29-s + (0.142 + 0.989i)31-s + (0.755 + 0.654i)33-s + (0.959 + 0.281i)37-s + (0.654 + 0.755i)39-s + (0.959 − 0.281i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.536 - 0.843i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.536 - 0.843i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1840\)    =    \(2^{4} \cdot 5 \cdot 23\)
Sign: $0.536 - 0.843i$
Analytic conductor: \(8.54492\)
Root analytic conductor: \(8.54492\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1840} (1067, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1840,\ (0:\ ),\ 0.536 - 0.843i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.098399096 - 0.6033374419i\)
\(L(\frac12)\) \(\approx\) \(1.098399096 - 0.6033374419i\)
\(L(1)\) \(\approx\) \(1.012436602 + 0.02455697790i\)
\(L(1)\) \(\approx\) \(1.012436602 + 0.02455697790i\)

Euler product

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

−20.17222252236919456285274150425, −19.45312606148781789975083653661, −18.572485144972391966575420013332, −18.25123773354898201529690389036, −17.37610976825644913081701088500, −16.88401590707662019508303134492, −15.81896769300873038864454769551, −14.885395033199123556762935386630, −14.41130042903495549758762794797, −13.51462385262284915778151063715, −12.78969320501041937071700890051, −12.09599860771493883339611676502, −11.31792785878126420495583181457, −10.95673848117019541432391809389, −9.3981290176940685323911083080, −8.985314526745869082995939816715, −7.956083439267052922922704718419, −7.44929562343763776215026417935, −6.25269875615768890781137783251, −6.08840630885384168909871309673, −4.727258199778463375704734274916, −4.1400684874368406287824413661, −2.62036825858218766399108128478, −1.98842939595512844026246375549, −1.25460019496141528671131474284, 0.4571188513207390902832367816, 1.6790463308928148183210110072, 2.957091269711755526159954460775, 3.83558746579090302305207398464, 4.38243695860806398996400770554, 5.32380149983370653016373498661, 6.090631406378572959862538326737, 6.96173030639143038572955897591, 8.233829929385060201176580959981, 8.608650661247312415284551003152, 9.4630835137077072290693518442, 10.59503030023695794987856648006, 10.90222592102842740157097259850, 11.44307621082882417068410658697, 12.61058552049691254886552502368, 13.52878644562649912516841817933, 14.18724125757366809386953164167, 14.97918794523417164436702222463, 15.5365675422792983375332762018, 16.42948496925203107766871410870, 17.05332288351732584429274357230, 17.62127761749030128299236728453, 18.42634856559222530761497165708, 19.62844788015548716767152883633, 20.03097749030297511403890716317

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