Properties

Label 1-97-97.41-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: $\chi_{97} (41, \cdot )$
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

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

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