Properties

Degree 1
Conductor 59
Sign $0.0904 - 0.995i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,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,χ)  = 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

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

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(59\)
\( \varepsilon \)  =  $0.0904 - 0.995i$
motivic weight  =  \(0\)
character  :  $\chi_{59} (23, \cdot )$
Sato-Tate  :  $\mu(58)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 59,\ (1:\ ),\ 0.0904 - 0.995i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.6308829600 - 0.5762041295i$
$L(\frac12,\chi)$  $\approx$  $0.6308829600 - 0.5762041295i$
$L(\chi,1)$  $\approx$  0.8518787471 + 0.01514123542i
$L(1,\chi)$  $\approx$  0.8518787471 + 0.01514123542i

Euler product

\[\begin{aligned} L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]
\[\begin{aligned} L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]

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