↑

Properties

Label 1-1580-1580.67-r0-0-0
Degree $1$
Conductor $1580$
Sign $0.931 + 0.364i$
Analytic cond. $7.33748$
Root an. cond. $7.33748$
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  + (0.663 − 0.748i)3-s + (−0.663 + 0.748i)7-s + (−0.120 − 0.992i)9-s + (−0.568 + 0.822i)11-s + (−0.464 − 0.885i)13-s + (0.464 + 0.885i)17-s + (−0.354 − 0.935i)19-s + (0.120 + 0.992i)21-s − i·23-s + (−0.822 − 0.568i)27-s + (−0.120 + 0.992i)29-s + (0.970 − 0.239i)31-s + (0.239 + 0.970i)33-s + (0.935 − 0.354i)37-s + (−0.970 − 0.239i)39-s + ⋯
L(s)  = 1  + (0.663 − 0.748i)3-s + (−0.663 + 0.748i)7-s + (−0.120 − 0.992i)9-s + (−0.568 + 0.822i)11-s + (−0.464 − 0.885i)13-s + (0.464 + 0.885i)17-s + (−0.354 − 0.935i)19-s + (0.120 + 0.992i)21-s − i·23-s + (−0.822 − 0.568i)27-s + (−0.120 + 0.992i)29-s + (0.970 − 0.239i)31-s + (0.239 + 0.970i)33-s + (0.935 − 0.354i)37-s + (−0.970 − 0.239i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1580 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.931 + 0.364i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1580 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.931 + 0.364i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1580\)    =    \(2^{2} \cdot 5 \cdot 79\)
Sign: $0.931 + 0.364i$
Analytic conductor: \(7.33748\)
Root analytic conductor: \(7.33748\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1580} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1580,\ (0:\ ),\ 0.931 + 0.364i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.508155869 + 0.2845530423i\)
\(L(\frac12)\) \(\approx\) \(1.508155869 + 0.2845530423i\)
\(L(1)\) \(\approx\) \(1.146764082 - 0.08113391296i\)
\(L(1)\) \(\approx\) \(1.146764082 - 0.08113391296i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
79 \( 1 \)
good3 \( 1 + (0.663 - 0.748i)T \)
7 \( 1 + (-0.663 + 0.748i)T \)
11 \( 1 + (-0.568 + 0.822i)T \)
13 \( 1 + (-0.464 - 0.885i)T \)
17 \( 1 + (0.464 + 0.885i)T \)
19 \( 1 + (-0.354 - 0.935i)T \)
23 \( 1 - iT \)
29 \( 1 + (-0.120 + 0.992i)T \)
31 \( 1 + (0.970 - 0.239i)T \)
37 \( 1 + (0.935 - 0.354i)T \)
41 \( 1 + (0.568 + 0.822i)T \)
43 \( 1 + (-0.822 + 0.568i)T \)
47 \( 1 + (0.935 + 0.354i)T \)
53 \( 1 + (0.663 + 0.748i)T \)
59 \( 1 + (0.885 + 0.464i)T \)
61 \( 1 + (-0.354 - 0.935i)T \)
67 \( 1 + (-0.239 + 0.970i)T \)
71 \( 1 + (0.748 - 0.663i)T \)
73 \( 1 + (0.464 - 0.885i)T \)
83 \( 1 + (0.464 + 0.885i)T \)
89 \( 1 + (0.748 + 0.663i)T \)
97 \( 1 + (0.935 - 0.354i)T \)
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

−20.60734808043607856657977899246, −19.74779305547375118476836215933, −18.97648158606778981350228606607, −18.61666367375702622277609994430, −17.14082981956437711068941857535, −16.52480675044859936481317498845, −16.14614693473919051366741544696, −15.244191057059159041114128971563, −14.29726923240951162193981621838, −13.85186765335435403252227805706, −13.16453573723262898528359003276, −12.10120656873397292847557168765, −11.20087055219472476092411241496, −10.23012703776765150950351293865, −9.94619478324541320897523789436, −8.98049831185224211249040509098, −8.19739465981973084667246470492, −7.42453179776005575970912332340, −6.4798597874040147914503954270, −5.48610623612808387323139770543, −4.48258879821735615780385881189, −3.8465076129392457758377353948, −2.94468452442367771880027282874, −2.19546740181369799539650586170, −0.58928002541429890927375347772, 1.0149561743942814014180561856, 2.24333946674979952492302656990, 2.77736138597656528372358613005, 3.65381151592956691100735446851, 4.910025315923831583975519769551, 5.83653938219786324337336898459, 6.60098620991315384394938177044, 7.537128102214747632860349404324, 8.06946852665460821663612972034, 9.05530421446792765694791012704, 9.673404856771116430947475863448, 10.506500955155508825147550405329, 11.691373152686021095139893982418, 12.52358854118955940962381664509, 12.89624219619998552235238468931, 13.55657084393711318017357400863, 14.80091100443815046678389499288, 15.10782583293775990959557812399, 15.808100288721905759122016939125, 16.993101357071891746514698867787, 17.81791810730857719054716512114, 18.25931418650677615374544035575, 19.2348877555183484014403155770, 19.67999367125609862269686594922, 20.312426176072597677691141465486

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