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Properties

Label 1-1840-1840.1459-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: $\chi_{1840} (1459, \cdot )$
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

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

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