L-function 1-760-760.93-r1-0-0

• Label: 1-760-760.93-r1-0-0
• Degree: $1$
• Conductor: $760$
• Sign: $-0.128 + 0.991i$
• Analytic cond.: $81.6733$
• Root an. cond.: $81.6733$
• Motivic weight: $0$
• Arithmetic: yes
• Rational: no
• Primitive: yes
• Self-dual: no
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

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

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 760 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.128 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$760$    =    $2^{3} \cdot 5 \cdot 19$
Sign:	$-0.128 + 0.991i$
Analytic conductor:	$81.6733$
Root analytic conductor:	$81.6733$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{760} (93, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 760,\ (1:\ ),\ -0.128 + 0.991i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$1.781259005 + 2.026741517i$
$L(1)$	$\approx$	$1.483536822 + 0.3234367420i$

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	$1 + (0.984 - 0.173i)T$
	7	$1 + (0.866 + 0.5i)T$
	11	$1 + (0.5 + 0.866i)T$
	13	$1 + (-0.984 - 0.173i)T$
	17	$1 + (-0.342 + 0.939i)T$
	23	$1 + (-0.642 + 0.766i)T$
	29	$1 + (-0.939 + 0.342i)T$
	31	$1 + (-0.5 + 0.866i)T$
	37	$1 - iT$

Imaginary part of the first few zeros on the critical line

−21.83920011611133695475249463851, −21.12255735800008018430364211795, −20.27418171137935138311695992178, −19.80403538932420933678379689357, −18.81364123253481077506389790778, −18.132967744244973447255663489341, −16.97492640538566668526352844363, −16.3714701296491687147381595763, −15.26907037026351412577042521016, −14.49200742017279617732106918942, −13.98960275910318914949277550542, −13.218175838094515179101400083245, −12.04306190682531580992719747684, −11.18108632504908017686445103180, −10.26527993204427299102638293040, −9.31304109439674505912837925072, −8.62508028614084094670444711668, −7.634294029960210149983302192431, −7.09203806852108384099335236361, −5.66678190972186508010457548057, −4.521191836562251976567355474623, −3.89356864554098551017601655131, −2.66783003359886126902030908101, −1.82033709824723855531403112363, −0.47627960924019079717962521333, 1.56226245122343193722145523952, 2.03294307177652210597763688951, 3.25528603680213563470792006992, 4.3092860566459810533199468862, 5.137641348364465306546884626812, 6.46384016465553926807171751098, 7.47367722722609043349042603401, 8.07463223285663939855154444061, 9.05211473895817424209002383148, 9.71461946376203592285965903178, 10.73039799166551004107915794245, 11.95305164426342824830350812352, 12.50384507683018997916711898053, 13.48773120850002216056001014902, 14.49365439546105607922174865125, 14.925037284362370221777671227734, 15.55218582980464195167148950949, 16.904798566118433672881420797541, 17.71607357523205241854544719776, 18.375110504115473080650016233758, 19.40415998124243060133913043038, 19.98841028209133509466696600945, 20.66518209357750326190405817737, 21.70860617192290510767942553652, 22.064978344350585166769971305269

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