Properties

Degree 1
Conductor 97
Sign $0.233 + 0.972i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,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,χ)  = 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

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

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(97\)
\( \varepsilon \)  =  $0.233 + 0.972i$
motivic weight  =  \(0\)
character  :  $\chi_{97} (13, \cdot )$
Sato-Tate  :  $\mu(96)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 97,\ (1:\ ),\ 0.233 + 0.972i)$
$L(\chi,\frac{1}{2})$  $\approx$  $1.529545784 + 1.206181387i$
$L(\frac12,\chi)$  $\approx$  $1.529545784 + 1.206181387i$
$L(\chi,1)$  $\approx$  1.354721858 + 0.2752485214i
$L(1,\chi)$  $\approx$  1.354721858 + 0.2752485214i

Euler product

\[\begin{aligned} L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]
\[\begin{aligned} L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]

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