Properties

Degree 1
Conductor 59
Sign $0.418 - 0.908i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.468 − 0.883i)2-s + (−0.725 + 0.687i)3-s + (−0.561 − 0.827i)4-s + (0.976 − 0.214i)5-s + (0.267 + 0.963i)6-s + (0.796 − 0.605i)7-s + (−0.994 + 0.108i)8-s + (0.0541 − 0.998i)9-s + (0.267 − 0.963i)10-s + (−0.370 − 0.928i)11-s + (0.976 + 0.214i)12-s + (0.0541 + 0.998i)13-s + (−0.161 − 0.986i)14-s + (−0.561 + 0.827i)15-s + (−0.370 + 0.928i)16-s + (0.796 + 0.605i)17-s + ⋯
L(s,χ)  = 1  + (0.468 − 0.883i)2-s + (−0.725 + 0.687i)3-s + (−0.561 − 0.827i)4-s + (0.976 − 0.214i)5-s + (0.267 + 0.963i)6-s + (0.796 − 0.605i)7-s + (−0.994 + 0.108i)8-s + (0.0541 − 0.998i)9-s + (0.267 − 0.963i)10-s + (−0.370 − 0.928i)11-s + (0.976 + 0.214i)12-s + (0.0541 + 0.998i)13-s + (−0.161 − 0.986i)14-s + (−0.561 + 0.827i)15-s + (−0.370 + 0.928i)16-s + (0.796 + 0.605i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 59 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.418 - 0.908i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 59 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.418 - 0.908i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(59\)
\( \varepsilon \)  =  $0.418 - 0.908i$
motivic weight  =  \(0\)
character  :  $\chi_{59} (45, \cdot )$
Sato-Tate  :  $\mu(29)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 59,\ (0:\ ),\ 0.418 - 0.908i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.8411635465 - 0.5388188788i$
$L(\frac12,\chi)$  $\approx$  $0.8411635465 - 0.5388188788i$
$L(\chi,1)$  $\approx$  1.020844500 - 0.4420549957i
$L(1,\chi)$  $\approx$  1.020844500 - 0.4420549957i

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

−33.2575062746461354158508892063, −31.956811070903998133436522944883, −30.521507156391790966595694329574, −29.9913930766757972747481697979, −28.444386342078422837039263407818, −27.40877362205850327193574550562, −25.577871830633581509893712164456, −25.097267187436618685871525190552, −23.96691939207904381449768416277, −22.86856731507236973525418367974, −21.944847050470777235802603802980, −20.803131440946918743237128042, −18.4286165093624821294107352030, −17.83783475386755150713917507550, −16.96796017585186260367033455896, −15.38263828234717042202691007163, −14.189804236323544366895193486968, −12.99152039603517926615368624726, −12.0337429467743609818567040346, −10.25465127337739513965423806245, −8.35871372638492356506367060435, −7.09690126324663242313127367869, −5.78085827627988416066127355341, −4.9848030493549688935576834879, −2.29255163264460012232051563762, 1.586431888081172259316806824304, 3.793321643404110851774743083736, 5.09362674323528931763442234242, 6.14054223138762419796636835558, 8.84669549738490506884254294590, 10.2195435783581645590492723455, 10.94275338738938515643468377138, 12.24531372882803286227899919420, 13.72235008084928664325228404609, 14.63248067838380441945639912542, 16.48111024210723590169701972988, 17.52760833650774287448635453050, 18.729131662990142154745412502963, 20.461667357268039142105261589534, 21.42210683993985828717668798060, 21.77034054109840993993568506008, 23.47242512400384664852676394335, 24.045469109481315270338946450971, 26.106162325651886128067102323919, 27.32117065217976685031764204682, 28.223816394134513694072568464112, 29.3389771838051822516781516388, 29.89282013935989708940463925750, 31.52585523399322142073428166215, 32.57319150393906530549097432061

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