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Properties

Label 1-97-97.13-r1-0-0
Degree $1$
Conductor $97$
Sign $0.233 + 0.972i$
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  + (0.608 − 0.793i)2-s + (0.130 + 0.991i)3-s + (−0.258 − 0.965i)4-s + (−0.0654 + 0.997i)5-s + (0.866 + 0.5i)6-s + (0.896 + 0.442i)7-s + (−0.923 − 0.382i)8-s + (−0.965 + 0.258i)9-s + (0.751 + 0.659i)10-s + (−0.793 + 0.608i)11-s + (0.923 − 0.382i)12-s + (−0.997 − 0.0654i)13-s + (0.896 − 0.442i)14-s + (−0.997 + 0.0654i)15-s + (−0.866 + 0.5i)16-s + (0.442 + 0.896i)17-s + ⋯
L(s)  = 1  + (0.608 − 0.793i)2-s + (0.130 + 0.991i)3-s + (−0.258 − 0.965i)4-s + (−0.0654 + 0.997i)5-s + (0.866 + 0.5i)6-s + (0.896 + 0.442i)7-s + (−0.923 − 0.382i)8-s + (−0.965 + 0.258i)9-s + (0.751 + 0.659i)10-s + (−0.793 + 0.608i)11-s + (0.923 − 0.382i)12-s + (−0.997 − 0.0654i)13-s + (0.896 − 0.442i)14-s + (−0.997 + 0.0654i)15-s + (−0.866 + 0.5i)16-s + (0.442 + 0.896i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 97 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.233 + 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 97 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.233 + 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(97\)
Sign: $0.233 + 0.972i$
Analytic conductor: \(10.4240\)
Root analytic conductor: \(10.4240\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{97} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 97,\ (1:\ ),\ 0.233 + 0.972i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.529545784 + 1.206181387i\)
\(L(\frac12)\) \(\approx\) \(1.529545784 + 1.206181387i\)
\(L(1)\) \(\approx\) \(1.354721858 + 0.2752485214i\)
\(L(1)\) \(\approx\) \(1.354721858 + 0.2752485214i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad97 \( 1 \)
good2 \( 1 + (0.608 - 0.793i)T \)
3 \( 1 + (0.130 + 0.991i)T \)
5 \( 1 + (-0.0654 + 0.997i)T \)
7 \( 1 + (0.896 + 0.442i)T \)
11 \( 1 + (-0.793 + 0.608i)T \)
13 \( 1 + (-0.997 - 0.0654i)T \)
17 \( 1 + (0.442 + 0.896i)T \)
19 \( 1 + (0.831 + 0.555i)T \)
23 \( 1 + (0.946 + 0.321i)T \)
29 \( 1 + (-0.751 + 0.659i)T \)
31 \( 1 + (0.991 - 0.130i)T \)
37 \( 1 + (-0.321 - 0.946i)T \)
41 \( 1 + (0.659 + 0.751i)T \)
43 \( 1 + (0.965 + 0.258i)T \)
47 \( 1 + (0.707 - 0.707i)T \)
53 \( 1 + (-0.793 - 0.608i)T \)
59 \( 1 + (-0.946 + 0.321i)T \)
61 \( 1 + (-0.5 - 0.866i)T \)
67 \( 1 + (0.555 - 0.831i)T \)
71 \( 1 + (0.659 - 0.751i)T \)
73 \( 1 + (-0.258 + 0.965i)T \)
79 \( 1 + (0.382 - 0.923i)T \)
83 \( 1 + (-0.896 + 0.442i)T \)
89 \( 1 + (0.923 + 0.382i)T \)
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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

−29.780357957305944145762943919535, −28.87624685827599311699420373568, −27.29528222365851020670794408515, −26.323869329550842036105619146792, −24.92191713724123788834605542827, −24.34732212435471267335732695269, −23.77417870534631886476994391522, −22.6528311256627722177306766257, −21.07038931922851310618014699358, −20.38359107446377488664986480130, −18.82448289213200568178162196640, −17.53288409651362028593690994, −16.90833422754556019110438906584, −15.588693086938930312462054093324, −14.16389721396309486019949595139, −13.49298326888822676792412667109, −12.40681836181625746652924853654, −11.4686286174920623003741994940, −9.093091454416495996309916253176, −7.92308418915151031132144429844, −7.307741091211754651955275453137, −5.57113748390261904982719789320, −4.71644775374173299957848643157, −2.74682330911218482113492033386, −0.699495166838195827506924087066, 2.20745007628904743352583622649, 3.31771715734725871586732941972, 4.73065202992696079635623046441, 5.68872180715199737843735589048, 7.71492473388655847404663150803, 9.457624123580063507858402651985, 10.43070328271705030543838288781, 11.22501107955230657511101644057, 12.40425147133683551243926267617, 14.13826052503343016297433583790, 14.82898683225956765711656520672, 15.51215088864588889281601746403, 17.439420903599889876418756235, 18.57261005874157891041576310609, 19.70926415026397790111798034912, 20.905915618386142619254205569561, 21.543567738379742033049477219649, 22.4914629943169438080659008492, 23.33986493936636231990842555331, 24.761160323901075903878558327082, 26.21900278985215669626883398211, 27.14324846506113402007266850301, 27.9856431258797512476739529752, 29.025091350616615986784021405097, 30.26152219733818279988588084534

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