L-function 1-760-760.347-r0-0-0

• Label: 1-760-760.347-r0-0-0
• Degree: $1$
• Conductor: $760$
• Sign: $0.829 + 0.558i$
• Analytic cond.: $3.52942$
• Root an. cond.: $3.52942$
• Motivic weight: $0$
• Arithmetic: yes
• Rational: no
• Primitive: yes
• Self-dual: no
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

+ (−0.342 + 0.939i)3^-s + (0.866 + 0.5i)7^-s + (−0.766 − 0.642i)9^-s + (−0.5 − 0.866i)11^-s + (−0.342 − 0.939i)13^-s + (0.642 + 0.766i)17^-s + (−0.766 + 0.642i)21^-s + (0.984 + 0.173i)23^-s + (0.866 − 0.5i)27^-s + (0.766 + 0.642i)29^-s + (0.5 − 0.866i)31^-s + (0.984 − 0.173i)33^-s − i·37^-s + 39^-s + (−0.939 − 0.342i)41^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 760 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.829 + 0.558i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$760$    =    $2^{3} \cdot 5 \cdot 19$
Sign:	$0.829 + 0.558i$
Analytic conductor:	$3.52942$
Root analytic conductor:	$3.52942$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{760} (347, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 760,\ (0:\ ),\ 0.829 + 0.558i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$1.289905603 + 0.3937540678i$
$L(1)$	$\approx$	$1.015605782 + 0.2421904685i$

Euler product

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

	$p$	$F_p(T)$
bad	2	$1$
	5	$1$
	19	$1$
good	3	$1 + (-0.342 + 0.939i)T$
	7	$1 + (0.866 + 0.5i)T$
	11	$1 + (-0.5 - 0.866i)T$
	13	$1 + (-0.342 - 0.939i)T$
	17	$1 + (0.642 + 0.766i)T$
	23	$1 + (0.984 + 0.173i)T$
	29	$1 + (0.766 + 0.642i)T$
	31	$1 + (0.5 - 0.866i)T$
	37	$1 - iT$

Imaginary part of the first few zeros on the critical line

−22.61281536549681763738976902738, −21.414808700004334433566880424509, −20.71279158267286107436816696955, −19.89127279964072642479893152771, −18.93193887570079774327087689897, −18.34349692368711108219787201494, −17.407174404451786502572875661639, −17.015109925643057246218001510899, −15.93411651697994262899023602739, −14.77722570824573750482849544296, −14.0455229880646553937283620307, −13.384867003531812316670169875456, −12.28535663721184707920386472793, −11.777357739618659211881333264165, −10.866182831884854646716044899985, −9.95869075138189034834049302916, −8.77080803509479286440510223673, −7.765896928147569036523097131525, −7.21752563671038936762361015403, −6.38690054618726679484511374367, −5.03388891170133040179722556078, −4.6223280633438256503889696831, −2.93305422299856142453508808212, −1.93292735798419926767731461961, −0.980132611952826404081949597706, 0.89487778318119672429697151283, 2.54565629074807208943664957027, 3.42341083852151022107866788759, 4.54832931505553352650666774171, 5.48623397781278747788239745676, 5.86271199278118082833538802555, 7.44090556819381895886243798027, 8.4190604251367658690868165494, 9.00539012318955579883051972427, 10.32449315735358664430252672327, 10.68678357550628898462635721458, 11.65903524031717465712611985882, 12.4169403738931580213513955215, 13.55890663835802837178848210585, 14.625476793104250217412925716191, 15.14468940885231639827558481171, 15.90573029643196541562551049249, 16.86740182995173161834362918391, 17.490160111504376982653760876942, 18.34227746563693558722312845461, 19.2764276042414133047125208403, 20.294683932661648596914773177363, 21.15413427965674949367279479851, 21.49692295597800004852338595402, 22.35653721674064865439944221136

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