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Properties

Label 1-59-59.23-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

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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: $\chi_{59} (23, \cdot )$
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 \( 1 + (0.856 + 0.515i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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