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Properties

Label 1-1840-1840.539-r0-0-0
Degree $1$
Conductor $1840$
Sign $0.940 + 0.339i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.940 + 0.339i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.940 + 0.339i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1840\)    =    \(2^{4} \cdot 5 \cdot 23\)
Sign: $0.940 + 0.339i$
Analytic conductor: \(8.54492\)
Root analytic conductor: \(8.54492\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1840} (539, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1840,\ (0:\ ),\ 0.940 + 0.339i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.713390117 + 0.4745746006i\)
\(L(\frac12)\) \(\approx\) \(2.713390117 + 0.4745746006i\)
\(L(1)\) \(\approx\) \(1.684110750 + 0.03780514280i\)
\(L(1)\) \(\approx\) \(1.684110750 + 0.03780514280i\)

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

−20.20529710677239798280191215203, −19.553021502907100476104918864543, −18.807876009487005816034530941493, −17.668941306101830958636587757912, −17.395175366986061868373659912664, −16.32060163865631582322945822045, −15.44304337835232984011110051327, −15.03408876657147718129661942435, −14.15643988551480091005420587399, −13.59046321232959829206743976855, −12.9517083058524178885625564566, −11.76200545396229327168251000218, −10.98907364499999240483707924651, −10.42065290960654444465573557907, −9.36851750387288816826557889568, −8.837066071597473689479639621117, −8.02846750480259243490521388877, −7.33374649869736284350122009016, −6.45617330427853578392417700674, −5.27286593671481889570695185148, −4.31859463348135283122747054940, −3.95282398305367867852265347922, −2.78445129447715797446519274433, −1.97249561013388538093056495367, −0.91077083618236256712822196791, 1.40807988930758701430519941900, 1.760725677065195329089337278, 2.82662333878382273091246257456, 3.90341144072752747241185461336, 4.40956881729168569854152296247, 5.71647407598084548358039014074, 6.540526200482448097438919843286, 7.25980174516330118864146175072, 8.23235848047151912058262392507, 8.926496825635888830136179007543, 9.16808210180836946026189204144, 10.50917916354339689286931819100, 11.270357283675229543738723719676, 12.163659224320565995770466767614, 12.65332663340729760085736811845, 13.80653168153706325541844230389, 14.2576135685240146524533131424, 14.78473325196060711771513063820, 15.67004760065980319604344683562, 16.41724219449155206643782103803, 17.514121408715895397726595436076, 18.007295950608093712584337544144, 18.90052712007851046236874191763, 19.31442480878311930618644786015, 20.22976825117680811206347875435

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