Properties

Label 1-97-97.71-r1-0-0
Degree $1$
Conductor $97$
Sign $0.590 - 0.807i$
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 + (0.751 + 0.659i)5-s + (0.866 − 0.5i)6-s + (−0.946 − 0.321i)7-s + (−0.382 − 0.923i)8-s + (0.965 + 0.258i)9-s + (0.997 + 0.0654i)10-s + (0.608 − 0.793i)11-s + (0.382 − 0.923i)12-s + (0.659 − 0.751i)13-s + (−0.946 + 0.321i)14-s + (0.659 + 0.751i)15-s + (−0.866 − 0.5i)16-s + (0.321 + 0.946i)17-s + ⋯
L(s)  = 1  + (0.793 − 0.608i)2-s + (0.991 + 0.130i)3-s + (0.258 − 0.965i)4-s + (0.751 + 0.659i)5-s + (0.866 − 0.5i)6-s + (−0.946 − 0.321i)7-s + (−0.382 − 0.923i)8-s + (0.965 + 0.258i)9-s + (0.997 + 0.0654i)10-s + (0.608 − 0.793i)11-s + (0.382 − 0.923i)12-s + (0.659 − 0.751i)13-s + (−0.946 + 0.321i)14-s + (0.659 + 0.751i)15-s + (−0.866 − 0.5i)16-s + (0.321 + 0.946i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 97 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.590 - 0.807i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 97 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.590 - 0.807i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(97\)
Sign: $0.590 - 0.807i$
Analytic conductor: \(10.4240\)
Root analytic conductor: \(10.4240\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{97} (71, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 97,\ (1:\ ),\ 0.590 - 0.807i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.454209077 - 1.753984969i\)
\(L(\frac12)\) \(\approx\) \(3.454209077 - 1.753984969i\)
\(L(1)\) \(\approx\) \(2.228674354 - 0.7756777578i\)
\(L(1)\) \(\approx\) \(2.228674354 - 0.7756777578i\)

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

−30.17846460328928066815875957696, −29.30541520581374512252747028584, −27.93151724756719563576255321583, −26.232161459238266357990093159206, −25.60692387688992112869542361683, −24.95828355694741876797513950888, −23.965645053912491523409012407141, −22.657483814075962521992038689799, −21.539556281114324486672968220318, −20.70565322762390437913351570209, −19.68455934325299440336157824985, −18.20657632956367251950646294831, −16.85193556330992154057775311750, −15.84993205454926373872115495532, −14.806482237961946108885528574067, −13.55898032488298724165522405024, −13.103360203376011190248554779447, −11.873284114013038249397389403004, −9.48821734175804028849711997376, −8.96681278554980027371166914766, −7.30783518254591754244175665655, −6.285445590080019663946810373279, −4.74899684320175157472876713468, −3.400908296560307154219529717843, −1.98989437248518768271366833645, 1.550610111891095577564044308432, 3.12179800773345495299914642274, 3.71222432583836681959863707596, 5.789456862248631579258293352931, 6.81189409604627957684400615254, 8.75756593049185130999077492560, 10.08122540406993215074379390588, 10.67192768333737630371232832048, 12.61019225696415648156129582271, 13.469072605800991019190329417120, 14.30354976679435714028534771062, 15.222717070487783072609116265137, 16.58438454900806044908171057240, 18.56862435499991041475565588848, 19.213684170104633485789497820702, 20.30997890676380236125310962693, 21.281347586747554960223958308282, 22.17222554927610601008401899511, 23.098149113403303078164679818159, 24.61394535308989145608049213235, 25.432903888938252471214269136114, 26.41991359158087471786825185023, 27.65591464156332900678654452836, 29.11017304722790836767631718262, 29.91052753042721794724015755288

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