↑

Properties

Label 1-1840-1840.629-r0-0-0
Degree $1$
Conductor $1840$
Sign $0.480 - 0.877i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.755 − 0.654i)3-s + (0.415 + 0.909i)7-s + (0.142 + 0.989i)9-s + (−0.281 − 0.959i)11-s + (0.909 + 0.415i)13-s + (−0.841 + 0.540i)17-s + (0.540 − 0.841i)19-s + (0.281 − 0.959i)21-s + (0.540 − 0.841i)27-s + (−0.540 − 0.841i)29-s + (−0.654 − 0.755i)31-s + (−0.415 + 0.909i)33-s + (−0.989 + 0.142i)37-s + (−0.415 − 0.909i)39-s + (0.142 − 0.989i)41-s + ⋯
L(s)  = 1  + (−0.755 − 0.654i)3-s + (0.415 + 0.909i)7-s + (0.142 + 0.989i)9-s + (−0.281 − 0.959i)11-s + (0.909 + 0.415i)13-s + (−0.841 + 0.540i)17-s + (0.540 − 0.841i)19-s + (0.281 − 0.959i)21-s + (0.540 − 0.841i)27-s + (−0.540 − 0.841i)29-s + (−0.654 − 0.755i)31-s + (−0.415 + 0.909i)33-s + (−0.989 + 0.142i)37-s + (−0.415 − 0.909i)39-s + (0.142 − 0.989i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.480 - 0.877i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.480 - 0.877i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1840\)    =    \(2^{4} \cdot 5 \cdot 23\)
Sign: $0.480 - 0.877i$
Analytic conductor: \(8.54492\)
Root analytic conductor: \(8.54492\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1840} (629, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1840,\ (0:\ ),\ 0.480 - 0.877i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9635915464 - 0.5710243049i\)
\(L(\frac12)\) \(\approx\) \(0.9635915464 - 0.5710243049i\)
\(L(1)\) \(\approx\) \(0.8547089405 - 0.1679301862i\)
\(L(1)\) \(\approx\) \(0.8547089405 - 0.1679301862i\)

Euler product

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

Imaginary part of the first few zeros on the critical line

−20.4457114723566672941848323488, −19.85883711822061120469897498916, −18.439282538059121401964127694379, −17.95173665459893038638476867903, −17.43208244381880318005316732102, −16.4771562277568579463748404488, −16.022823072382469158146490151697, −15.20481872622128140915748099512, −14.470003331552649661538474618970, −13.612010820675034543196421074957, −12.75287203111402584490527958288, −12.015898105566946446024434556443, −11.02931620788107318916602053940, −10.70292879175778175419762999490, −9.89202535826188362688101045212, −9.13599776603385424379533594474, −8.10934946233658411754008398520, −7.17385452689452942256646537397, −6.57407792382836736483839755082, −5.416620812297887492047126686394, −4.916168958427440187093018171048, −3.97274268564288013695707179683, −3.382234318504925812855448492041, −1.87379069052124150359172029000, −0.90163163698634254337791750065, 0.55878321346650788269724494061, 1.750062799683683183761232425097, 2.430047221731183670185506707364, 3.64681266453964760686192716465, 4.73572195763864748662705853965, 5.61080127069613959716240137057, 6.07809663690626807223692293830, 6.92242858626904958273201616174, 7.910122453260872837675484437493, 8.598654412979773883066515296159, 9.2933531495428323556666563711, 10.61532864014087811201188571521, 11.26264588717690346606774881305, 11.60685159370687482301709530441, 12.58270125280810269291693129270, 13.369616899583178433984756469277, 13.79923080938272387338373320185, 14.96256822555986031712445921847, 15.764895514192868862708009243, 16.29962201456597989036491766559, 17.25270075649804769673687537312, 17.8891576573359248230009456189, 18.53248305116042965461319277208, 19.05221771916385392528008444942, 19.810529091761059337560461194233

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