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Properties

Label 1-97-97.60-r1-0-0
Degree $1$
Conductor $97$
Sign $0.947 - 0.320i$
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 + (−0.946 + 0.321i)5-s + (−0.866 − 0.5i)6-s + (0.751 − 0.659i)7-s + (−0.382 − 0.923i)8-s + (−0.258 − 0.965i)9-s + (−0.442 − 0.896i)10-s + (−0.991 − 0.130i)11-s + (0.382 − 0.923i)12-s + (0.321 + 0.946i)13-s + (0.751 + 0.659i)14-s + (0.321 − 0.946i)15-s + (0.866 − 0.5i)16-s + (0.659 − 0.751i)17-s + ⋯
L(s)  = 1  + (0.130 + 0.991i)2-s + (−0.608 + 0.793i)3-s + (−0.965 + 0.258i)4-s + (−0.946 + 0.321i)5-s + (−0.866 − 0.5i)6-s + (0.751 − 0.659i)7-s + (−0.382 − 0.923i)8-s + (−0.258 − 0.965i)9-s + (−0.442 − 0.896i)10-s + (−0.991 − 0.130i)11-s + (0.382 − 0.923i)12-s + (0.321 + 0.946i)13-s + (0.751 + 0.659i)14-s + (0.321 − 0.946i)15-s + (0.866 − 0.5i)16-s + (0.659 − 0.751i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 97 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.947 - 0.320i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 97 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.947 - 0.320i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(97\)
Sign: $0.947 - 0.320i$
Analytic conductor: \(10.4240\)
Root analytic conductor: \(10.4240\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{97} (60, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 97,\ (1:\ ),\ 0.947 - 0.320i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4312926023 - 0.07092173387i\)
\(L(\frac12)\) \(\approx\) \(0.4312926023 - 0.07092173387i\)
\(L(1)\) \(\approx\) \(0.5278904103 + 0.3471297493i\)
\(L(1)\) \(\approx\) \(0.5278904103 + 0.3471297493i\)

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 \( 1 + (-0.946 + 0.321i)T \)
7 \( 1 + (0.751 - 0.659i)T \)
11 \( 1 + (-0.991 - 0.130i)T \)
13 \( 1 + (0.321 + 0.946i)T \)
17 \( 1 + (0.659 - 0.751i)T \)
19 \( 1 + (-0.195 + 0.980i)T \)
23 \( 1 + (-0.997 + 0.0654i)T \)
29 \( 1 + (0.442 - 0.896i)T \)
31 \( 1 + (-0.793 - 0.608i)T \)
37 \( 1 + (0.0654 - 0.997i)T \)
41 \( 1 + (0.896 + 0.442i)T \)
43 \( 1 + (0.258 - 0.965i)T \)
47 \( 1 + (-0.707 - 0.707i)T \)
53 \( 1 + (-0.991 + 0.130i)T \)
59 \( 1 + (0.997 + 0.0654i)T \)
61 \( 1 + (-0.5 - 0.866i)T \)
67 \( 1 + (-0.980 - 0.195i)T \)
71 \( 1 + (0.896 - 0.442i)T \)
73 \( 1 + (-0.965 - 0.258i)T \)
79 \( 1 + (-0.923 + 0.382i)T \)
83 \( 1 + (-0.751 - 0.659i)T \)
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

−30.05229272925444675456533371226, −28.74282813597497212021578061864, −27.982809924129612375529430722177, −27.4432363188510479109474609464, −25.75843239318444998396634155142, −24.082319640505446365059118966947, −23.67827296072419498633804152929, −22.5891518296732342581381493774, −21.46319200998194095513436073353, −20.26890921410413686356002323258, −19.29821505468100282498940309522, −18.27465531508234146404233236315, −17.61285961458782445988850924840, −15.91654422335561451436308547881, −14.59046492989233749171737977583, −12.997836184111320621443522953173, −12.41550543725545316794740383278, −11.36361725368271978258485722579, −10.53955788742438636001822292400, −8.50966778027091733881796326707, −7.7954974139198776200229683523, −5.653654197878378495206702500691, −4.70408748084294776609337870002, −2.858588230376336673651147141468, −1.28874745598585977103595170019, 0.22899888934307631735437797184, 3.729387092546796125488648700472, 4.55541230618800458353401005, 5.837131202152510462943322117323, 7.31722634129365948638288236039, 8.25697286055330421541311559528, 9.85446151962359129241735838925, 11.06641444464082201343680996732, 12.191909197031565630603496962664, 13.96953883900046380413184741978, 14.84243249155001103517349426601, 16.02803588118973964560452985014, 16.50868953083326820168918386251, 17.85890995409633717313646924842, 18.77813237529109907545670909777, 20.60418806478095086356405924191, 21.5244868447509143449918343349, 22.91297563223167781258550132634, 23.40055404544315094868724845882, 24.207680756689450783672266030634, 25.95754463464224596872186204154, 26.72863600637502662136309859990, 27.3674005436260199502133760057, 28.33471867211230178087178311756, 29.912475612261187697731814364955

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