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Properties

Label 1-97-97.23-r1-0-0
Degree $1$
Conductor $97$
Sign $0.404 + 0.914i$
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  + (−0.130 + 0.991i)2-s + (0.608 + 0.793i)3-s + (−0.965 − 0.258i)4-s + (0.321 − 0.946i)5-s + (−0.866 + 0.5i)6-s + (0.659 − 0.751i)7-s + (0.382 − 0.923i)8-s + (−0.258 + 0.965i)9-s + (0.896 + 0.442i)10-s + (0.991 − 0.130i)11-s + (−0.382 − 0.923i)12-s + (0.946 + 0.321i)13-s + (0.659 + 0.751i)14-s + (0.946 − 0.321i)15-s + (0.866 + 0.5i)16-s + (−0.751 + 0.659i)17-s + ⋯
L(s)  = 1  + (−0.130 + 0.991i)2-s + (0.608 + 0.793i)3-s + (−0.965 − 0.258i)4-s + (0.321 − 0.946i)5-s + (−0.866 + 0.5i)6-s + (0.659 − 0.751i)7-s + (0.382 − 0.923i)8-s + (−0.258 + 0.965i)9-s + (0.896 + 0.442i)10-s + (0.991 − 0.130i)11-s + (−0.382 − 0.923i)12-s + (0.946 + 0.321i)13-s + (0.659 + 0.751i)14-s + (0.946 − 0.321i)15-s + (0.866 + 0.5i)16-s + (−0.751 + 0.659i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 97 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.404 + 0.914i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 97 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.404 + 0.914i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(97\)
Sign: $0.404 + 0.914i$
Analytic conductor: \(10.4240\)
Root analytic conductor: \(10.4240\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{97} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 97,\ (1:\ ),\ 0.404 + 0.914i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.782851231 + 1.160885470i\)
\(L(\frac12)\) \(\approx\) \(1.782851231 + 1.160885470i\)
\(L(1)\) \(\approx\) \(1.222134935 + 0.6317093364i\)
\(L(1)\) \(\approx\) \(1.222134935 + 0.6317093364i\)

Euler product

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

−29.82780341441154197011757071779, −28.88068978889254995179815804926, −27.63295227617248852432607990079, −26.63002171434581420841311658074, −25.542675822402090242780303659775, −24.616782758268943259872267433142, −23.0952443144256339725009380635, −22.188775263622861629341434652852, −21.06494591914134672015015486364, −20.05636775078728681236240486656, −18.9728954664883110100722828788, −18.15790092578951937411523327047, −17.588081850142862078370636501631, −15.22351458814624320264276108997, −14.12508590967449341985672318039, −13.42383894461116570170210720349, −11.90313187158717171308485277773, −11.26066989726686212345910025627, −9.59692414659246097104471405464, −8.6641695568007038536417958263, −7.33499068001009755258254054491, −5.76899604241724416001299514184, −3.68824275797015580316197097045, −2.509486383843658485024978740574, −1.374607153179556242951776901012, 1.2520159374244747970118717633, 3.996721200494449740234074645922, 4.70410640799947593036216048047, 6.19695158594891376923627638332, 7.90116419139265405703440704511, 8.80151385608624812322090756421, 9.64700227238350419183450529175, 11.10875251516698363209058682876, 13.19054691089393943290618585692, 13.992883378983600183757232131784, 14.947771951353126669893491854, 16.282242538349268358106627937925, 16.808057238007911709518257718129, 17.9642518880162638751807381940, 19.61491024010848449934431816857, 20.53119932399725101122404504302, 21.62196415727980480832185518358, 22.77452426231541616784681157838, 24.206451391856323064320532639117, 24.72945415428123514630758784831, 25.97366439090387325752693361886, 26.73269629624377589808506790508, 27.75014509522513133588106335392, 28.45890924214028536735448167022, 30.420193982202584269438164053254

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