↑

Properties

Label 1-1588-1588.39-r0-0-0
Degree $1$
Conductor $1588$
Sign $0.810 - 0.585i$
Analytic cond. $7.37464$
Root an. cond. $7.37464$
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  + (−0.916 − 0.400i)3-s + (−0.220 − 0.975i)5-s + (−0.734 − 0.678i)7-s + (0.678 + 0.734i)9-s + (−0.950 − 0.312i)11-s + (0.429 + 0.902i)13-s + (−0.189 + 0.981i)15-s + (0.998 + 0.0475i)17-s + (0.987 − 0.158i)19-s + (0.400 + 0.916i)21-s + (−0.204 − 0.978i)23-s + (−0.902 + 0.429i)25-s + (−0.327 − 0.945i)27-s + (−0.605 + 0.795i)29-s + (0.959 + 0.281i)31-s + ⋯
L(s)  = 1  + (−0.916 − 0.400i)3-s + (−0.220 − 0.975i)5-s + (−0.734 − 0.678i)7-s + (0.678 + 0.734i)9-s + (−0.950 − 0.312i)11-s + (0.429 + 0.902i)13-s + (−0.189 + 0.981i)15-s + (0.998 + 0.0475i)17-s + (0.987 − 0.158i)19-s + (0.400 + 0.916i)21-s + (−0.204 − 0.978i)23-s + (−0.902 + 0.429i)25-s + (−0.327 − 0.945i)27-s + (−0.605 + 0.795i)29-s + (0.959 + 0.281i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1588 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.810 - 0.585i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1588 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.810 - 0.585i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1588\)    =    \(2^{2} \cdot 397\)
Sign: $0.810 - 0.585i$
Analytic conductor: \(7.37464\)
Root analytic conductor: \(7.37464\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1588} (39, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1588,\ (0:\ ),\ 0.810 - 0.585i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8227308280 - 0.2660525040i\)
\(L(\frac12)\) \(\approx\) \(0.8227308280 - 0.2660525040i\)
\(L(1)\) \(\approx\) \(0.6969474205 - 0.2000745786i\)
\(L(1)\) \(\approx\) \(0.6969474205 - 0.2000745786i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
397 \( 1 \)
good3 \( 1 + (-0.916 - 0.400i)T \)
5 \( 1 + (-0.220 - 0.975i)T \)
7 \( 1 + (-0.734 - 0.678i)T \)
11 \( 1 + (-0.950 - 0.312i)T \)
13 \( 1 + (0.429 + 0.902i)T \)
17 \( 1 + (0.998 + 0.0475i)T \)
19 \( 1 + (0.987 - 0.158i)T \)
23 \( 1 + (-0.204 - 0.978i)T \)
29 \( 1 + (-0.605 + 0.795i)T \)
31 \( 1 + (0.959 + 0.281i)T \)
37 \( 1 + (-0.444 + 0.895i)T \)
41 \( 1 + (0.342 - 0.939i)T \)
43 \( 1 + (0.0475 + 0.998i)T \)
47 \( 1 + (0.266 + 0.963i)T \)
53 \( 1 + (0.189 + 0.981i)T \)
59 \( 1 + (0.993 + 0.110i)T \)
61 \( 1 + (-0.429 + 0.902i)T \)
67 \( 1 + (-0.997 + 0.0634i)T \)
71 \( 1 + (0.690 + 0.723i)T \)
73 \( 1 + (0.873 + 0.486i)T \)
79 \( 1 + (0.939 - 0.342i)T \)
83 \( 1 + (-0.888 - 0.458i)T \)
89 \( 1 + (0.996 - 0.0792i)T \)
97 \( 1 + (-0.386 + 0.922i)T \)
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

−20.840861627473662249421511701361, −19.70667018503820119796145233343, −18.84788386676728894705065498050, −18.24231296801494512384491300506, −17.82051888266837203828807111397, −16.782980404453972244435355937759, −15.80785753579052071911229864808, −15.56900906269507287200992793981, −14.89475057646435598757270737459, −13.70949444318498385761633799411, −12.91525863599553557544137463087, −12.046853088579983742005612569840, −11.53034611571674204766303541507, −10.55711376811289519284055905896, −10.02030680641841355639648126756, −9.45099528730847948521944309144, −7.999368347502397976631893980003, −7.39404112439868985376622731053, −6.395693502440626658329038579365, −5.64823499737254866938369102022, −5.212599074695335911101125199726, −3.699792131784707777073143535479, −3.26496565382143779555344683164, −2.183800262371748337302809727492, −0.580651794819667337174545100168, 0.74340432999773850851903911115, 1.41471644352254489166442855358, 2.85850927491644982940377998116, 3.97809200183755118192725049335, 4.7843469416817988241027389631, 5.55853581334746658426013566820, 6.330243964567567228243296488647, 7.269625580993175771269829539704, 7.89745261468763226094454926084, 8.89446750802703294623608806905, 9.86942974242273809408672955671, 10.52423134161298713085928738717, 11.40947170979856753481612029981, 12.20789882296090866249491228883, 12.7491347967824683863629412382, 13.51838317485955331222474238865, 14.06867577914285754660714582011, 15.60595617616089759761508877299, 16.259009018108337891345733229698, 16.5108692316071535455779438875, 17.27797418744266019157802105882, 18.2367436147953554121014015434, 18.89803686059544088071134474634, 19.522017460775866142835913203, 20.58445424952261976235625472640

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