L-function 1-760-760.477-r0-0-0

• Label: 1-760-760.477-r0-0-0
• Degree: $1$
• Conductor: $760$
• Sign: $0.647 - 0.761i$
• 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

+ (0.984 − 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

$\begin{aligned}\Lambda(s)=\mathstrut & 760 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.647 - 0.761i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$760$    =    $2^{3} \cdot 5 \cdot 19$
Sign:	$0.647 - 0.761i$
Analytic conductor:	$3.52942$
Root analytic conductor:	$3.52942$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{760} (477, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 760,\ (0:\ ),\ 0.647 - 0.761i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$1.707391250 - 0.7891048687i$
$L(1)$	$\approx$	$1.365387019 - 0.2606391198i$

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

−22.23760294365039637491206466475, −21.62910080796862116617845645975, −21.109942070549255252062377259872, −19.70289021328343531586000173896, −19.48503037807927859188411663521, −18.88061714201814557953919828887, −17.69844766744494803256744208300, −16.656921492065540311893090647870, −15.9639328410182328379085273958, −15.111584498545641078404361633638, −14.39583243017335722291091529683, −13.60018202491063714757118629876, −12.705626824237458677829920859275, −12.02104192446084660593516237721, −10.7176718077541779185787547024, −9.86282463038976561714686676512, −9.09877237196255624137363027514, −8.45930332208025823345943470472, −7.38986759006058321816246854640, −6.49836193678197838795235669652, −5.4516453262065514207941763224, −4.235413847452786350883939606171, −3.27513606332793737063827490342, −2.63418777134192913398839051980, −1.32987700477728807267241349526, 0.85458386458687140817515351589, 2.30052214832615045472808023134, 3.01254071218297330102638009119, 4.09418778838614091953870922216, 4.92166208225579520099438842376, 6.536596320167783242796394372334, 7.09403608445824013570862150677, 7.90430627948123506659854887157, 9.03615566875532957271138283873, 9.7966975950608953081810373523, 10.24668863059223975969893956264, 11.82631669208403381208853651213, 12.53510305259271174180984116382, 13.33150343452099502616991755091, 14.12909569535971725559674305778, 14.8902675766640278454281689232, 15.636482654070217952353539105055, 16.62686445904138063935559070400, 17.414367292006148519084680912420, 18.4728760148509880576727593225, 19.25284587795709138130425443269, 19.89286180321896628847645970242, 20.49067836035152987511127151266, 21.32016231473094290764637832919, 22.53498844443815685836482165147

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