Properties

Degree 1
Conductor 97
Sign $-0.453 + 0.891i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.793 − 0.608i)2-s + (−0.991 + 0.130i)3-s + (0.258 + 0.965i)4-s + (0.659 + 0.751i)5-s + (0.866 + 0.5i)6-s + (0.321 + 0.946i)7-s + (0.382 − 0.923i)8-s + (0.965 − 0.258i)9-s + (−0.0654 − 0.997i)10-s + (−0.608 − 0.793i)11-s + (−0.382 − 0.923i)12-s + (−0.751 + 0.659i)13-s + (0.321 − 0.946i)14-s + (−0.751 − 0.659i)15-s + (−0.866 + 0.5i)16-s + (0.946 + 0.321i)17-s + ⋯
L(s,χ)  = 1  + (−0.793 − 0.608i)2-s + (−0.991 + 0.130i)3-s + (0.258 + 0.965i)4-s + (0.659 + 0.751i)5-s + (0.866 + 0.5i)6-s + (0.321 + 0.946i)7-s + (0.382 − 0.923i)8-s + (0.965 − 0.258i)9-s + (−0.0654 − 0.997i)10-s + (−0.608 − 0.793i)11-s + (−0.382 − 0.923i)12-s + (−0.751 + 0.659i)13-s + (0.321 − 0.946i)14-s + (−0.751 − 0.659i)15-s + (−0.866 + 0.5i)16-s + (0.946 + 0.321i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 97 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (-0.453 + 0.891i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 97 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (-0.453 + 0.891i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(97\)
\( \varepsilon \)  =  $-0.453 + 0.891i$
motivic weight  =  \(0\)
character  :  $\chi_{97} (29, \cdot )$
Sato-Tate  :  $\mu(96)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 97,\ (1:\ ),\ -0.453 + 0.891i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.3020827704 + 0.4926416505i$
$L(\frac12,\chi)$  $\approx$  $0.3020827704 + 0.4926416505i$
$L(\chi,1)$  $\approx$  0.5594733506 + 0.1091387486i
$L(1,\chi)$  $\approx$  0.5594733506 + 0.1091387486i

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

−29.27828052168694604888977111105, −28.31378382345498377525243003993, −27.57503360353535299273122967943, −26.49587145254320678105879645, −25.23526401649619930189717071020, −24.29985877264970379929214418224, −23.520679119165318680214799782186, −22.46570072049287061007384055999, −20.808223477840656214557226715712, −20.00263601261396331138127887541, −18.28895030926658825335156664610, −17.65920181663957453202665020852, −16.78294100782956065326940907504, −16.0465064164454590571904014726, −14.488510446573152930585746042629, −13.152169935709626142006419402, −11.85652084983131241445068630578, −10.22085923217673683574485109150, −9.91222027725438759120352113719, −7.95989882486960864966756037354, −7.067827802283290090459246995037, −5.54858312942825766273642920709, −4.83849844416533665645213833651, −1.66216573965821128702346629914, −0.38650160665344208583924969131, 1.67582158441943925685635578775, 3.14270600138717148878833002263, 5.22583575555680402396391335735, 6.45611227221377645689708567127, 7.865522188080520987598802380473, 9.5084978736375945558341548805, 10.31556483553493850697358825556, 11.49051858737815244679529904931, 12.17972750724113683067122387779, 13.73768567215380948817032705965, 15.4321876024078460571877242744, 16.57254988663294932772488029898, 17.634843326669089864755714322585, 18.4276239901637135441252153983, 19.11971531350570340727354500676, 21.05802825192534277517780772512, 21.68455687090331287011611679660, 22.34042174804799135808230461450, 23.9787243342599615825678584397, 25.13932954861860701554453556257, 26.382811440055689214709231834772, 27.138850361895162088047797926012, 28.33957887748844874536149132411, 28.97638606287938626830840999437, 29.810107402629425366700009989661

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