Properties

Label 1-1840-1840.1019-r0-0-0
Degree $1$
Conductor $1840$
Sign $0.425 - 0.905i$
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  + (−0.909 − 0.415i)3-s + (0.841 − 0.540i)7-s + (0.654 + 0.755i)9-s + (−0.989 − 0.142i)11-s + (−0.540 + 0.841i)13-s + (−0.959 − 0.281i)17-s + (0.281 + 0.959i)19-s + (−0.989 + 0.142i)21-s + (−0.281 − 0.959i)27-s + (−0.281 + 0.959i)29-s + (−0.415 − 0.909i)31-s + (0.841 + 0.540i)33-s + (0.755 − 0.654i)37-s + (0.841 − 0.540i)39-s + (0.654 − 0.755i)41-s + ⋯
L(s)  = 1  + (−0.909 − 0.415i)3-s + (0.841 − 0.540i)7-s + (0.654 + 0.755i)9-s + (−0.989 − 0.142i)11-s + (−0.540 + 0.841i)13-s + (−0.959 − 0.281i)17-s + (0.281 + 0.959i)19-s + (−0.989 + 0.142i)21-s + (−0.281 − 0.959i)27-s + (−0.281 + 0.959i)29-s + (−0.415 − 0.909i)31-s + (0.841 + 0.540i)33-s + (0.755 − 0.654i)37-s + (0.841 − 0.540i)39-s + (0.654 − 0.755i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.425 - 0.905i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.425 - 0.905i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1840\)    =    \(2^{4} \cdot 5 \cdot 23\)
Sign: $0.425 - 0.905i$
Analytic conductor: \(8.54492\)
Root analytic conductor: \(8.54492\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1840} (1019, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1840,\ (0:\ ),\ 0.425 - 0.905i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7821336292 - 0.4967543415i\)
\(L(\frac12)\) \(\approx\) \(0.7821336292 - 0.4967543415i\)
\(L(1)\) \(\approx\) \(0.7678892858 - 0.1449302696i\)
\(L(1)\) \(\approx\) \(0.7678892858 - 0.1449302696i\)

Euler product

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

−20.41534663971660908206948664703, −19.626817746549089870283505163296, −18.47416830594131234317070251197, −17.90947882733922751065647242407, −17.55313743146648422291257062460, −16.6635896281977610979052457131, −15.68463483443633892901755607154, −15.308627593397046161927693155162, −14.6809013955569181292036706071, −13.37068795759859551296442098197, −12.83428271863739814461554365861, −11.92883217450588630722205500829, −11.2745982175580580135284208622, −10.65927592077910989356327124450, −9.91098308891097731176554662312, −9.03133151141851934172276772285, −8.10945501384820029204814926347, −7.34306665960234460040740458159, −6.36894544161207383835672216708, −5.45877014324666601293736951820, −4.96314017261622754279992443030, −4.27633969654303160885158773357, −2.949493270727321063414483057656, −2.12036733884761949983947030896, −0.809875508404939312745356918178, 0.4956622314701392331322149733, 1.72041309245923035698909132671, 2.35222502599338185295511583774, 3.88894957428719568082459548533, 4.66667003385922889467814708952, 5.34613272890358150740838336315, 6.14912605951960400587240623349, 7.29446759999696514300862938011, 7.487284024773911043811063748224, 8.54515365592062230276749025952, 9.59435615435167115976595583715, 10.5736176141896961574920093739, 10.99723525396752589496528862984, 11.76396705724792163499160894140, 12.49325938767170618384271823799, 13.336258891181164787944763254, 13.9570343164986882953102221303, 14.80989032269146502140844826568, 15.773865217598123057711828118977, 16.54083512764402247272004760231, 17.028558576098548761413270706791, 17.92497517034040449653217785237, 18.36883167811856857417679728871, 19.05650308463155116041235188546, 20.1029702062857704528445925163

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