Properties

Degree 1
Conductor 103
Sign $0.488 - 0.872i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.932 − 0.361i)2-s + (0.982 − 0.183i)3-s + (0.739 − 0.673i)4-s + (0.273 − 0.961i)5-s + (0.850 − 0.526i)6-s + (0.445 + 0.895i)7-s + (0.445 − 0.895i)8-s + (0.932 − 0.361i)9-s + (−0.0922 − 0.995i)10-s + (−0.932 + 0.361i)11-s + (0.602 − 0.798i)12-s + (0.445 + 0.895i)13-s + (0.739 + 0.673i)14-s + (0.0922 − 0.995i)15-s + (0.0922 − 0.995i)16-s + (−0.850 − 0.526i)17-s + ⋯
L(s,χ)  = 1  + (0.932 − 0.361i)2-s + (0.982 − 0.183i)3-s + (0.739 − 0.673i)4-s + (0.273 − 0.961i)5-s + (0.850 − 0.526i)6-s + (0.445 + 0.895i)7-s + (0.445 − 0.895i)8-s + (0.932 − 0.361i)9-s + (−0.0922 − 0.995i)10-s + (−0.932 + 0.361i)11-s + (0.602 − 0.798i)12-s + (0.445 + 0.895i)13-s + (0.739 + 0.673i)14-s + (0.0922 − 0.995i)15-s + (0.0922 − 0.995i)16-s + (−0.850 − 0.526i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 103 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (0.488 - 0.872i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 103 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (0.488 - 0.872i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(103\)
\( \varepsilon \)  =  $0.488 - 0.872i$
motivic weight  =  \(0\)
character  :  $\chi_{103} (95, \cdot )$
Sato-Tate  :  $\mu(34)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 103,\ (1:\ ),\ 0.488 - 0.872i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(3.741821139 - 2.193156005i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(3.741821139 - 2.193156005i\)
\(L(\chi,1)\)  \(\approx\)  \(2.415025531 - 0.9322737261i\)
\(L(1,\chi)\)  \(\approx\)  \(2.415025531 - 0.9322737261i\)

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

−30.22470176845100634284253236163, −29.20011176112323897632406596356, −27.18522038766431192229253371530, −26.299871454286723178966803039560, −25.69954821134529558585148962869, −24.54667308964857201565218252052, −23.50176953995320118146634102029, −22.48390703831645934175081128434, −21.27014768801408234099555284395, −20.68213799486370733032262546804, −19.47248774511792900313590747468, −18.10697194248888742287809514015, −16.78108284261021617583338595639, −15.233068607675929677868382991023, −14.90896478923168508931224849142, −13.47726516912265093671647755445, −13.223114140858714326304553711384, −11.02595832556578814788395831011, −10.36954562602760387957159947220, −8.29818764281386199360949845707, −7.46998895663047541219163482467, −6.1939969664742360179107667712, −4.50631618813258781103648488930, −3.34406656860176918089848416974, −2.23920962390575267029740136362, 1.63564710303189983593245386860, 2.58584440104751164009465177427, 4.308576968763614557953363347469, 5.29898974316014201745107253644, 6.91669605134385535170852017569, 8.53056649241422930496327079935, 9.4251028595097256591860409426, 11.069963753907893176001677890746, 12.49989357349876919583325942157, 13.108578064496900752370370946580, 14.19973730151944827131287836486, 15.300895209053241316666796635158, 16.11870387514883929698848776909, 18.04082121257150042617742580325, 19.17135408762608032184169715575, 20.23482833754332646052515102702, 21.167954773821447696774858150, 21.54321200801825621546753878285, 23.397369889141989230522622326, 24.22804755961787300866360102636, 25.026073211454972140430124891574, 25.87555414860516539655610541215, 27.57320872433585535729795303926, 28.62472107881165124862399243884, 29.40215468107066797011290364461

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