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Properties

Label 1-1840-1840.1347-r0-0-0
Degree $1$
Conductor $1840$
Sign $0.727 - 0.685i$
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  + (−0.415 − 0.909i)3-s + (−0.540 + 0.841i)7-s + (−0.654 + 0.755i)9-s + (0.989 − 0.142i)11-s + (0.841 − 0.540i)13-s + (−0.281 − 0.959i)17-s + (0.281 − 0.959i)19-s + (0.989 + 0.142i)21-s + (0.959 + 0.281i)27-s + (0.281 + 0.959i)29-s + (−0.415 + 0.909i)31-s + (−0.540 − 0.841i)33-s + (−0.654 + 0.755i)37-s + (−0.841 − 0.540i)39-s + (0.654 + 0.755i)41-s + ⋯
L(s)  = 1  + (−0.415 − 0.909i)3-s + (−0.540 + 0.841i)7-s + (−0.654 + 0.755i)9-s + (0.989 − 0.142i)11-s + (0.841 − 0.540i)13-s + (−0.281 − 0.959i)17-s + (0.281 − 0.959i)19-s + (0.989 + 0.142i)21-s + (0.959 + 0.281i)27-s + (0.281 + 0.959i)29-s + (−0.415 + 0.909i)31-s + (−0.540 − 0.841i)33-s + (−0.654 + 0.755i)37-s + (−0.841 − 0.540i)39-s + (0.654 + 0.755i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.727 - 0.685i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.727 - 0.685i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1840\)    =    \(2^{4} \cdot 5 \cdot 23\)
Sign: $0.727 - 0.685i$
Analytic conductor: \(8.54492\)
Root analytic conductor: \(8.54492\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1840} (1347, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1840,\ (0:\ ),\ 0.727 - 0.685i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.228410944 - 0.4874737600i\)
\(L(\frac12)\) \(\approx\) \(1.228410944 - 0.4874737600i\)
\(L(1)\) \(\approx\) \(0.9321394397 - 0.2200725948i\)
\(L(1)\) \(\approx\) \(0.9321394397 - 0.2200725948i\)

Euler product

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

−20.43262984160232829647641911834, −19.51561760142372440399132437888, −18.93939453095566377825585759939, −17.76077962918415946446978993321, −17.11283735126331893418001180663, −16.62800866129403088574630336732, −15.90848647556198338401594508777, −15.21491270255587424741106107652, −14.2489721342833812331929730077, −13.82048609388027317742903015001, −12.667409832344871407238982077860, −11.99055310530279206077646607246, −11.02562136995314609381685805449, −10.61790141009235708118244029759, −9.652595906422499478136205350708, −9.19187135500522742520146571949, −8.224463473717649508190963045174, −7.157873573418819901097903177567, −6.189190554820876825355725685243, −5.88432396418485939658157210488, −4.45701750747293735453127868859, −3.924416907801447761887683427015, −3.45574211289302847037322559139, −1.928718049603248800929062992425, −0.79393296668201340958379140746, 0.73304121348823961777642912676, 1.6529500619945629438604964951, 2.7670396578477582074919479131, 3.39014915301050828631502335581, 4.85271220164426602984981051679, 5.50587175231138499768108784341, 6.52977345914996806766500111110, 6.74707666984982210167490908755, 7.87984050132316802740414658890, 8.80780457682945605291459905174, 9.23437286553423502349008125201, 10.45079668511081862031782794146, 11.392885190102111443819712422229, 11.77900375071509279465560845990, 12.68566448994729261662421006578, 13.22704472110373769494607459163, 14.020667616610802483356044179070, 14.80138995137439805120208728, 15.95524176150344860610056311114, 16.20195900120498010246270771570, 17.33867765706992914168371887975, 17.98301511489903296873114498633, 18.44782515996606212759304850849, 19.34613939629421972469101656455, 19.8102572847497846254489409961

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