L-function 1-693-693.128-r0-0-0

• Label: 1-693-693.128-r0-0-0
• Degree: $1$
• Conductor: $693$
• Sign: $0.647 + 0.762i$
• Analytic cond.: $3.21827$
• Root an. cond.: $3.21827$
• Motivic weight: $0$
• Arithmetic: yes
• Rational: no
• Primitive: yes
• Self-dual: no
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

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

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.647 + 0.762i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$693$    =    $3^{2} \cdot 7 \cdot 11$
Sign:	$0.647 + 0.762i$
Analytic conductor:	$3.21827$
Root analytic conductor:	$3.21827$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{693} (128, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 693,\ (0:\ ),\ 0.647 + 0.762i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$0.7970334262 + 0.3686792142i$
$L(1)$	$\approx$	$0.7215744209 + 0.1130824716i$

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	$1 + (-0.978 - 0.207i)T$
	5	$1 + (-0.309 + 0.951i)T$
	13	$1 + (0.978 + 0.207i)T$
	17	$1 + (0.669 + 0.743i)T$
	19	$1 + (0.104 - 0.994i)T$
	23	$1 - T$
	29	$1 + (0.913 + 0.406i)T$
	31	$1 + (0.669 - 0.743i)T$
	37	$1 + (0.913 + 0.406i)T$

Imaginary part of the first few zeros on the critical line

−22.91040087833409292164620938769, −21.26350477670723492781131362400, −20.84762421648043526514743973709, −19.99465141394682620721300632315, −19.35073692794420459093164737171, −18.32764216939497799331947996027, −17.78861969684702629992354318299, −16.63938333267784798181452007565, −16.20884562332381255168962839285, −15.5529206228838396069986982953, −14.41813471094808619127548360526, −13.47744202977028760163784746745, −12.19979615433958230432040179519, −11.79599120576081144700975182485, −10.60951797366134672922514221089, −9.81723158781421898197955460333, −8.91893663243235905172277903003, −8.15250643356348174413659193376, −7.55145899921482547689340565699, −6.21952839184760263897435579089, −5.53577329195478681152024130432, −4.27509374319073033095823838193, −3.05635463759805387883318057546, −1.658078241876461990188394226963, −0.74004967966945731329510277618, 1.06771303235602550972732484236, 2.35911376563722210313330806003, 3.23520099630358055576976767394, 4.20889789092859948397137456779, 6.0346753851905906950946954229, 6.5487048804976368416074753153, 7.67983072840065075210912935136, 8.24939709513090772164437317025, 9.36551407599616945333828242657, 10.22368101874516011273401622630, 10.987578012027614437813271678043, 11.603766823362824778890332868259, 12.568521833865965154190266771054, 13.73600909148196369943279700068, 14.7260631810002262957304903726, 15.59906223523618077422212152929, 16.191439406151567991456166926216, 17.27542809438656451962904044598, 18.0587190745759740839021981752, 18.630555750715682931141683496, 19.4664864101689741551467724749, 20.04507412894952047949060229289, 21.20906998087592852957183813823, 21.70687300837094211447097340220, 22.7806525414376651317804979343

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