L-function 1-760-760.597-r0-0-0

• Label: 1-760-760.597-r0-0-0
• Degree: $1$
• Conductor: $760$
• Sign: $-0.127 + 0.991i$
• 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

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

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 760 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.127 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$760$    =    $2^{3} \cdot 5 \cdot 19$
Sign:	$-0.127 + 0.991i$
Analytic conductor:	$3.52942$
Root analytic conductor:	$3.52942$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{760} (597, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 760,\ (0:\ ),\ -0.127 + 0.991i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$0.6275337586 + 0.7133807137i$
$L(1)$	$\approx$	$0.7741679747 + 0.2818989114i$

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	$1 + (-0.866 + 0.5i)T$
	7	$1 + iT$
	11	$1 - T$
	13	$1 + (0.866 + 0.5i)T$
	17	$1 + (0.866 - 0.5i)T$
	23	$1 + (0.866 + 0.5i)T$
	29	$1 + (0.5 - 0.866i)T$
	31	$1 - T$
	37	$1 - iT$

Imaginary part of the first few zeros on the critical line

−22.48569395902260504827004788440, −21.26458886798218771271399333552, −20.75882835923239409809913225558, −19.65680893854232745199424122983, −18.87125737930351496012998462766, −18.006001028674953841324824935011, −17.499072075213929120395756334496, −16.372377381120511962798618484029, −16.134002916290645815829309983731, −14.7938370783304680781080820680, −13.833188741695778461681444174400, −12.90070658744977391272193366798, −12.61272662357446956589562888693, −11.18531244467815918094811901634, −10.71865085523505972507200344712, −10.065026487966611314934089428607, −8.587218358752409438806400335446, −7.64060440535154382668679506957, −7.04996940971531596309562662144, −5.93161254976708452824951616037, −5.24072510355336029865548248171, −4.15891436124570254238445141278, −3.03281557118886712257514179344, −1.58715762442926598054583659303, −0.59531212268847674630530410959, 1.1572853253853268206204291732, 2.59849995832265760459346504029, 3.63804188135377230059892546405, 4.85187435323649471225070585608, 5.524098900669566841934619032184, 6.25190949240040548769190620371, 7.37257747796021771388347696221, 8.505805736035451347444217089194, 9.395173961033108886962980934675, 10.18345142040733668354635304474, 11.1847402820935994663795420943, 11.73701741168822553427971889516, 12.64314539817910023651642947556, 13.46977103614081860126366099151, 14.721401767115154712750931567403, 15.496210610844355743966295360894, 16.088878042964664828772946057760, 16.82646842140323932014923485199, 17.92108686751366071374221093299, 18.45265888444419193523604184764, 19.12893554185350659060178731920, 20.586742813355696901180364592376, 21.19797616079492700312792080773, 21.66605042961259898392943800614, 22.724738646457475840324919737403

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