L-function 1-693-693.563-r1-0-0

• Label: 1-693-693.563-r1-0-0
• Degree: $1$
• Conductor: $693$
• Sign: $0.927 + 0.374i$
• Analytic cond.: $74.4731$
• Root an. cond.: $74.4731$
• Motivic weight: $0$
• Arithmetic: yes
• Rational: no
• Primitive: yes
• Self-dual: no
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

+ (−0.104 + 0.994i)2^-s + (−0.978 − 0.207i)4^-s + (−0.809 + 0.587i)5^-s + (0.309 − 0.951i)8^-s + (−0.5 − 0.866i)10^-s + (−0.104 + 0.994i)13^-s + (0.913 + 0.406i)16^-s + (−0.913 − 0.406i)17^-s + (0.669 + 0.743i)19^-s + (0.913 − 0.406i)20^-s − 23^-s + (0.309 − 0.951i)25^-s + (−0.978 − 0.207i)26^-s + (−0.978 − 0.207i)29^-s + (−0.913 + 0.406i)31^-s + (−0.5 + 0.866i)32^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.927 + 0.374i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$693$    =    $3^{2} \cdot 7 \cdot 11$
Sign:	$0.927 + 0.374i$
Analytic conductor:	$74.4731$
Root analytic conductor:	$74.4731$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{693} (563, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 693,\ (1:\ ),\ 0.927 + 0.374i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$0.6194734808 + 0.1204278945i$
$L(1)$	$\approx$	$0.5648890763 + 0.3595325358i$

Euler product

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

	$p$	$F_p(T)$
bad	3	$1$
	7	$1$
	11	$1$
good	2	$1 + (-0.104 + 0.994i)T$
	5	$1 + (-0.809 + 0.587i)T$
	13	$1 + (-0.104 + 0.994i)T$
	17	$1 + (-0.913 - 0.406i)T$
	19	$1 + (0.669 + 0.743i)T$
	23	$1 - T$
	29	$1 + (-0.978 - 0.207i)T$
	31	$1 + (-0.913 + 0.406i)T$
	37	$1 + (-0.978 - 0.207i)T$

Imaginary part of the first few zeros on the critical line

−22.39178161202903480487029474411, −21.5713387280558948666720945420, −20.50121519579315409581603603143, −20.01009587642773277181007708811, −19.49985670856876096488544434169, −18.39825451566595407106700663019, −17.72580013681063335844518004718, −16.842371526476374156377461273715, −15.8192068738777668580697750796, −15.03080967251451753117193861168, −13.90289524606391750559312636670, −12.917346851776724875328855124564, −12.51717989965578993545387840849, −11.40811703645581857586963791420, −10.94187879394615247605168855905, −9.783106636803038363897747170, −8.9731513294663778007745819012, −8.13204138412962034983862850661, −7.37039545756577733990286795150, −5.73437842692206677505443643168, −4.803235776200112595542704047541, −3.94236738675638819443986213457, −3.06507672400665985787057273233, −1.84344292731751150479591124462, −0.63709967114583135469204994203, 0.25223876837190570275740490872, 1.9437500824212572848281594021, 3.559515717498308237716549575584, 4.20906691386139500665752129136, 5.31972835536866836622983998059, 6.40178956487103294022148633375, 7.15968846268368004031669005024, 7.83340579951311592781842185721, 8.83925346435263836162895824094, 9.65301867562708336132482237956, 10.72975125033870469244860536850, 11.67553624274738657802060713879, 12.584334642178556163312831083260, 13.80120291189280919788986448109, 14.3362931831381459859336008533, 15.177173762705939079826959058804, 16.05430240261731167363866711951, 16.45304042862562713365345063920, 17.69366267636794959437445788845, 18.327678681335776752181025212906, 19.091660291153192786411274264, 19.81388683226739065631726320186, 20.96150942518565460440078310081, 22.3206294182717058396209474292, 22.38764911109133205020578441825

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