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Properties

Label 1-97-97.87-r1-0-0
Degree $1$
Conductor $97$
Sign $-0.453 - 0.891i$
Analytic cond. $10.4240$
Root an. cond. $10.4240$
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  + (−0.793 + 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)  = 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

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

Invariants

Degree: \(1\)
Conductor: \(97\)
Sign: $-0.453 - 0.891i$
Analytic conductor: \(10.4240\)
Root analytic conductor: \(10.4240\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{97} (87, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 97,\ (1:\ ),\ -0.453 - 0.891i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3020827704 - 0.4926416505i\)
\(L(\frac12)\) \(\approx\) \(0.3020827704 - 0.4926416505i\)
\(L(1)\) \(\approx\) \(0.5594733506 - 0.1091387486i\)
\(L(1)\) \(\approx\) \(0.5594733506 - 0.1091387486i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad97 \( 1 \)
good2 \( 1 + (-0.793 + 0.608i)T \)
3 \( 1 + (-0.991 - 0.130i)T \)
5 \( 1 + (0.659 - 0.751i)T \)
7 \( 1 + (0.321 - 0.946i)T \)
11 \( 1 + (-0.608 + 0.793i)T \)
13 \( 1 + (-0.751 - 0.659i)T \)
17 \( 1 + (0.946 - 0.321i)T \)
19 \( 1 + (0.980 + 0.195i)T \)
23 \( 1 + (-0.896 - 0.442i)T \)
29 \( 1 + (0.0654 + 0.997i)T \)
31 \( 1 + (0.130 - 0.991i)T \)
37 \( 1 + (-0.442 - 0.896i)T \)
41 \( 1 + (-0.997 + 0.0654i)T \)
43 \( 1 + (-0.965 + 0.258i)T \)
47 \( 1 + (-0.707 - 0.707i)T \)
53 \( 1 + (-0.608 - 0.793i)T \)
59 \( 1 + (0.896 - 0.442i)T \)
61 \( 1 + (-0.5 + 0.866i)T \)
67 \( 1 + (-0.195 + 0.980i)T \)
71 \( 1 + (-0.997 - 0.0654i)T \)
73 \( 1 + (0.258 + 0.965i)T \)
79 \( 1 + (0.923 - 0.382i)T \)
83 \( 1 + (-0.321 - 0.946i)T \)
89 \( 1 + (-0.382 - 0.923i)T \)
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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

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

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