L-function 1-97-97.13-r1-0-0

• 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: $0$

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 + ⋯

Functional equation

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

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

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

Euler product

   $L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$

	$p$	$F_p(T)$
bad	97	$1$
good	2	$1 + (0.608 - 0.793i)T$
	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	$1 + (0.946 + 0.321i)T$

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