L-function 1-2793-2793.1376-r0-0-0

• Label: 1-2793-2793.1376-r0-0-0
• Degree: $1$
• Conductor: $2793$
• Sign: $0.326 - 0.945i$
• Analytic cond.: $12.9706$
• Root an. cond.: $12.9706$
• Motivic weight: $0$
• Arithmetic: yes
• Rational: no
• Primitive: yes
• Self-dual: no
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

+ (0.623 − 0.781i)2^-s + (−0.222 − 0.974i)4^-s + (0.900 + 0.433i)5^-s + (−0.900 − 0.433i)8^-s + (0.900 − 0.433i)10^-s + (0.988 + 0.149i)11^-s + (−0.365 − 0.930i)13^-s + (−0.900 + 0.433i)16^-s + (0.733 + 0.680i)17^-s + (0.222 − 0.974i)20^-s + (0.733 − 0.680i)22^-s + (0.733 − 0.680i)23^-s + (0.623 + 0.781i)25^-s + (−0.955 − 0.294i)26^-s + (−0.733 − 0.680i)29^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 2793 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.326 - 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$2793$    =    $3 \cdot 7^{2} \cdot 19$
Sign:	$0.326 - 0.945i$
Analytic conductor:	$12.9706$
Root analytic conductor:	$12.9706$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{2793} (1376, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 2793,\ (0:\ ),\ 0.326 - 0.945i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$2.498123575 - 1.779323198i$
$L(1)$	$\approx$	$1.609292363 - 0.7625010912i$

Euler product

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

	$p$	$F_p(T)$
bad	3	$1$
	7	$1$
	19	$1$
good	2	$1 + (0.623 - 0.781i)T$
	5	$1 + (0.900 + 0.433i)T$
	11	$1 + (0.988 + 0.149i)T$
	13	$1 + (-0.365 - 0.930i)T$
	17	$1 + (0.733 + 0.680i)T$
	23	$1 + (0.733 - 0.680i)T$
	29	$1 + (-0.733 - 0.680i)T$
	31	$1 + (0.5 + 0.866i)T$
	37	$1 + (0.733 + 0.680i)T$

Imaginary part of the first few zeros on the critical line

−19.247392896050778947244599815232, −18.54489404150253041778992003263, −17.631392899505023858018459564459, −17.04288311039053987994062046563, −16.63054533520752188400050766902, −15.96145133395643468220244209868, −14.92596237905908266591005368262, −14.36779258926640716317707580633, −13.78821397028589110763056976270, −13.22354272889937035226967600629, −12.27752643690385875508630414552, −11.852880543458264764482006138636, −10.90777500997062250915339405680, −9.637389020392057715653583446942, −9.19731543008163946920798024895, −8.64617383401568341859519606955, −7.32852910335378611914806726331, −7.0881156557816297042471044263, −5.875929566084849752120846613911, −5.67504279024934024118299291901, −4.60421389844155727609761181197, −4.02951874669967629729611378221, −2.982067661968992174064394423412, −2.08908720834858128617212116973, −1.00020833587783520949925397836, 0.95822139585253080851457387069, 1.66627070206080217306095897186, 2.6773664979679196339036052755, 3.20513383452719964698941761074, 4.19378766433262612890827155353, 5.016212303223876446110274718462, 5.86503836738130050998763790355, 6.33474642769757684753648802899, 7.23777929333949186252150061890, 8.39910208545301750113115019028, 9.306648467495962030343017386918, 9.939076824113328678734730158646, 10.44702372294225170363628658674, 11.24637990466229301356809482470, 11.97739744365129531427502232956, 12.890719670056732291920196096577, 13.1450341083973775356347186141, 14.27675261746566433186976171825, 14.59323961198725847397299525666, 15.14138638285907297471304287919, 16.245340729204150646981112346089, 17.35750828966269785580824733091, 17.521778617025489421082146789773, 18.68448703588520329110801941706, 19.02634746750532615853899954872

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