L-function 1-760-760.443-r0-0-0

• Label: 1-760-760.443-r0-0-0
• Degree: $1$
• Conductor: $760$
• Sign: $-0.460 + 0.887i$
• 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

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

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 760 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.460 + 0.887i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	\(1\)
Conductor:	\(760\)    =    \(2^{3} \cdot 5 \cdot 19\)
Sign:	$-0.460 + 0.887i$
Analytic conductor:	\(3.52942\)
Root analytic conductor:	\(3.52942\)
Motivic weight:	\(0\)
Rational:	no
Arithmetic:	yes
Character:	$\chi_{760} (443, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	\(0\)
Selberg data:	\((1,\ 760,\ (0:\ ),\ -0.460 + 0.887i)\)

Particular Values

\(L(\frac{1}{2})\)	\(\approx\)	\(0.4134321972 + 0.6801680654i\)
\(L(1)\)	\(\approx\)	\(0.7576383149 + 0.2568738252i\)

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

Imaginary part of the first few zeros on the critical line

−22.11575928865536331248729699458, −21.50296932871845535942479973730, −20.643134015466349221856039416012, −19.40288507219837781355952870230, −18.925434423561648488224372732678, −18.028876946500708767150003255516, −17.48040863199519879441210894146, −16.565093522072225380269111532106, −15.79054768439123103652835125394, −14.65594306525382571132853564046, −13.90532916058386091546218235635, −13.04246028445459096383920598733, −12.19177652388555389920768253825, −11.33869627622453243329802513084, −10.938767328691461962834889337511, −9.63986485877194683240462844117, −8.415452883921464585989392769229, −7.94155362370984090045339831714, −6.77430553199828026753483071637, −6.0227855306820711035423358429, −5.09383655291551403442024802420, −4.27306089886836964473669721795, −2.510839547839773874658873538707, −1.93814389626036392091066916048, −0.422841236686348237670871056774, 1.25740245021139799766955510478, 2.63899648986183781940425559505, 3.879174047194732012323292715924, 4.88000353738630575968331980856, 5.205899717535006215371456801863, 6.58153631464902577322243342097, 7.4564217475766663131675173838, 8.43208066452110896280895559490, 9.5720584549426314261438181822, 10.30714209828863551180844981593, 10.9682341508132892374608389629, 11.83730271295928907469290818967, 12.66127979603714317214080918324, 13.73100513448362566474021602362, 14.77404004103721040189053146496, 15.33867572806124537779954877865, 16.1179920850621920738490634257, 17.21683597966165589515765773194, 17.66971060457041640929557639546, 18.241082294841189741087177253212, 19.85275461245167187826454502605, 20.2589131499033481853837983450, 21.13776478749200477790896468250, 21.86400520391079985495350722780, 22.648150593608323453870831503604

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