↑

Properties

Label 1-1840-1840.613-r0-0-0
Degree $1$
Conductor $1840$
Sign $0.947 + 0.319i$
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  + (−0.654 + 0.755i)3-s + (0.909 − 0.415i)7-s + (−0.142 − 0.989i)9-s + (0.281 + 0.959i)11-s + (0.415 − 0.909i)13-s + (0.540 + 0.841i)17-s + (0.540 − 0.841i)19-s + (−0.281 + 0.959i)21-s + (0.841 + 0.540i)27-s + (0.540 + 0.841i)29-s + (−0.654 − 0.755i)31-s + (−0.909 − 0.415i)33-s + (0.142 + 0.989i)37-s + (0.415 + 0.909i)39-s + (0.142 − 0.989i)41-s + ⋯
L(s)  = 1  + (−0.654 + 0.755i)3-s + (0.909 − 0.415i)7-s + (−0.142 − 0.989i)9-s + (0.281 + 0.959i)11-s + (0.415 − 0.909i)13-s + (0.540 + 0.841i)17-s + (0.540 − 0.841i)19-s + (−0.281 + 0.959i)21-s + (0.841 + 0.540i)27-s + (0.540 + 0.841i)29-s + (−0.654 − 0.755i)31-s + (−0.909 − 0.415i)33-s + (0.142 + 0.989i)37-s + (0.415 + 0.909i)39-s + (0.142 − 0.989i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.947 + 0.319i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.947 + 0.319i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1840\)    =    \(2^{4} \cdot 5 \cdot 23\)
Sign: $0.947 + 0.319i$
Analytic conductor: \(8.54492\)
Root analytic conductor: \(8.54492\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1840} (613, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1840,\ (0:\ ),\ 0.947 + 0.319i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.522047389 + 0.2497033074i\)
\(L(\frac12)\) \(\approx\) \(1.522047389 + 0.2497033074i\)
\(L(1)\) \(\approx\) \(1.037651693 + 0.1615392637i\)
\(L(1)\) \(\approx\) \(1.037651693 + 0.1615392637i\)

Euler product

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

−19.9893345793645368648206918639, −19.12002644115713839799722588737, −18.54243635860243687978177791068, −18.07013205080282503830610335790, −17.246433800416089065429292795189, −16.34696286132427547664683390353, −16.10197903603651273301897884325, −14.722960270639168014418427066389, −14.02381467039371566875455681978, −13.6209828550812669727225601195, −12.46501633494981817063039561371, −11.79134766212403074968849064118, −11.38413966350269623401228906647, −10.63624966337065196099685008981, −9.50615089579738406603367090163, −8.58866726568786680284027377938, −7.93899840594883632304689320742, −7.14705185035097308230863531195, −6.22549706720153659086167396040, −5.58408979498914559969402256337, −4.85789192170924945098343639797, −3.78074451229325167445188164428, −2.62178242178594881225615717774, −1.64366215365850585223129058523, −0.92405520487875700272908203272, 0.81577437954677447939253589142, 1.76430787103335278230712494938, 3.16320803916852924760411351239, 3.96553013045215017251211131453, 4.81515761010944166736519990399, 5.3408392505973452840428122179, 6.3123153438539767493632806782, 7.217405850013236689549893369154, 8.05072979762669337853724364982, 8.945666907140556372135401694896, 9.84186027993739052135186896217, 10.51154331346045383474692382490, 11.078931609391989721530803271210, 11.90664615898791873525394793245, 12.57851146936959337720834552112, 13.533843771118023699740898118408, 14.53052856228195479438921064546, 15.10921125806823309988297257112, 15.62789587593584075128604527758, 16.726449688070301908722485467602, 17.16951686394015479508632376878, 17.92700246753743442071153636522, 18.33382831647523029533254318712, 19.75864673862949372338395477303, 20.31460663038695296464801581002

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