↑

Properties

Label 1-59-59.18-r1-0-0
Degree $1$
Conductor $59$
Sign $0.0904 + 0.995i$
Analytic cond. $6.34043$
Root an. cond. $6.34043$
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  + (−0.0541 − 0.998i)2-s + (0.907 + 0.419i)3-s + (−0.994 + 0.108i)4-s + (−0.947 + 0.319i)5-s + (0.370 − 0.928i)6-s + (−0.561 + 0.827i)7-s + (0.161 + 0.986i)8-s + (0.647 + 0.762i)9-s + (0.370 + 0.928i)10-s + (−0.976 − 0.214i)11-s + (−0.947 − 0.319i)12-s + (−0.647 + 0.762i)13-s + (0.856 + 0.515i)14-s + (−0.994 − 0.108i)15-s + (0.976 − 0.214i)16-s + (−0.561 − 0.827i)17-s + ⋯
L(s)  = 1  + (−0.0541 − 0.998i)2-s + (0.907 + 0.419i)3-s + (−0.994 + 0.108i)4-s + (−0.947 + 0.319i)5-s + (0.370 − 0.928i)6-s + (−0.561 + 0.827i)7-s + (0.161 + 0.986i)8-s + (0.647 + 0.762i)9-s + (0.370 + 0.928i)10-s + (−0.976 − 0.214i)11-s + (−0.947 − 0.319i)12-s + (−0.647 + 0.762i)13-s + (0.856 + 0.515i)14-s + (−0.994 − 0.108i)15-s + (0.976 − 0.214i)16-s + (−0.561 − 0.827i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 59 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0904 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 59 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0904 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(59\)
Sign: $0.0904 + 0.995i$
Analytic conductor: \(6.34043\)
Root analytic conductor: \(6.34043\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{59} (18, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 59,\ (1:\ ),\ 0.0904 + 0.995i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6308829600 + 0.5762041295i\)
\(L(\frac12)\) \(\approx\) \(0.6308829600 + 0.5762041295i\)
\(L(1)\) \(\approx\) \(0.8518787471 + 0.01514123542i\)
\(L(1)\) \(\approx\) \(0.8518787471 + 0.01514123542i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad59 \( 1 \)
good2 \( 1 + (-0.0541 - 0.998i)T \)
3 \( 1 + (0.907 + 0.419i)T \)
5 \( 1 + (-0.947 + 0.319i)T \)
7 \( 1 + (-0.561 + 0.827i)T \)
11 \( 1 + (-0.976 - 0.214i)T \)
13 \( 1 + (-0.647 + 0.762i)T \)
17 \( 1 + (-0.561 - 0.827i)T \)
19 \( 1 + (0.468 + 0.883i)T \)
23 \( 1 + (0.725 + 0.687i)T \)
29 \( 1 + (0.0541 - 0.998i)T \)
31 \( 1 + (-0.468 + 0.883i)T \)
37 \( 1 + (0.161 - 0.986i)T \)
41 \( 1 + (-0.725 + 0.687i)T \)
43 \( 1 + (-0.976 + 0.214i)T \)
47 \( 1 + (0.947 + 0.319i)T \)
53 \( 1 + (-0.370 + 0.928i)T \)
61 \( 1 + (-0.0541 - 0.998i)T \)
67 \( 1 + (0.161 + 0.986i)T \)
71 \( 1 + (-0.947 - 0.319i)T \)
73 \( 1 + (0.856 + 0.515i)T \)
79 \( 1 + (0.907 - 0.419i)T \)
83 \( 1 + (-0.267 + 0.963i)T \)
89 \( 1 + (-0.0541 + 0.998i)T \)
97 \( 1 + (0.856 - 0.515i)T \)
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

−32.22669596417399822692916580451, −31.312369995361316627540809497498, −30.413332333118757439040285652826, −28.75083773368781805740259179367, −27.17861631389851135069505658588, −26.445299922190884335302220730367, −25.53950395512133936297112592624, −24.15053912614690583628799777511, −23.660183894562650971748401261482, −22.347722265880935759307893055367, −20.38453048441063358979593352923, −19.54373841876328749776050454815, −18.38648023715737709451884093097, −16.979016576245886674329361563272, −15.63273830532051632990447734704, −14.92852300497031261644704406892, −13.34504941202935469423286066055, −12.71397865825657085203931267090, −10.29052845693793440807044538140, −8.79945617438069716547498767529, −7.71211584248925318486559243159, −6.920036970870814637224929221933, −4.75791264499370532056017266745, −3.3350743771768605892270519072, −0.41968598880887302916397962991, 2.45118532757447509008683204432, 3.45013378217214169936387332824, 4.92115661498489414340074233716, 7.58614089492498300044696833766, 8.848378920892471061430283049065, 9.90316021493664509333237921461, 11.30674448883856995887741080357, 12.52529094487446144375249791611, 13.841587032628307966801238496712, 15.13313273807353021257073886795, 16.21230251391056890022570173077, 18.44774295835445223353321919581, 19.106221343927574512090275457083, 20.05503559414774442279167510158, 21.2377985906587244316618083174, 22.177248207596064262350483042623, 23.38961122679651612173861335383, 25.03074008555771010736361496938, 26.57058457185994096178178369347, 26.9185541060848275615176706069, 28.29455055304434421365827552000, 29.341012326281007862361598932804, 30.8621102701231639127011422142, 31.49992593087427705971580788592, 31.97429397743551653267134243428

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