L-function 1-297-297.65-r0-0-0

• Label: 1-297-297.65-r0-0-0
• Degree: $1$
• Conductor: $297$
• Sign: $-0.597 + 0.802i$
• Analytic cond.: $1.37926$
• Root an. cond.: $1.37926$
• Motivic weight: $0$
• Arithmetic: yes
• Rational: no
• Primitive: yes
• Self-dual: no
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

+ (0.173 + 0.984i)2^-s + (−0.939 + 0.342i)4^-s + (−0.766 − 0.642i)5^-s + (0.939 + 0.342i)7^-s + (−0.5 − 0.866i)8^-s + (0.5 − 0.866i)10^-s + (−0.173 + 0.984i)13^-s + (−0.173 + 0.984i)14^-s + (0.766 − 0.642i)16^-s + (−0.5 + 0.866i)17^-s + (0.5 + 0.866i)19^-s + (0.939 + 0.342i)20^-s + (0.939 − 0.342i)23^-s + (0.173 + 0.984i)25^-s − 26^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 297 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.597 + 0.802i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$297$    =    $3^{3} \cdot 11$
Sign:	$-0.597 + 0.802i$
Analytic conductor:	$1.37926$
Root analytic conductor:	$1.37926$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{297} (65, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 297,\ (0:\ ),\ -0.597 + 0.802i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$0.4587163435 + 0.9133793361i$
$L(1)$	$\approx$	$0.7862892570 + 0.5574088528i$

Euler product

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

	$p$	$F_p(T)$
bad	3	$1$
	11	$1$
good	2	$1 + (0.173 + 0.984i)T$
	5	$1 + (-0.766 - 0.642i)T$
	7	$1 + (0.939 + 0.342i)T$
	13	$1 + (-0.173 + 0.984i)T$
	17	$1 + (-0.5 + 0.866i)T$
	19	$1 + (0.5 + 0.866i)T$
	23	$1 + (0.939 - 0.342i)T$
	29	$1 + (0.173 + 0.984i)T$
	31	$1 + (-0.939 + 0.342i)T$

Imaginary part of the first few zeros on the critical line

−24.942475812422333777386371452742, −23.90051298189912970964429771449, −23.10297666132545233804484754420, −22.394071188995239089456155380, −21.48428485035038936728090187516, −20.36461427089637121423157470849, −19.92528354970941096919080823599, −18.82230873815305724202806738837, −18.01427996653594824516352891152, −17.28946959675244297927268660909, −15.61630340935116848636812582007, −14.85477918390741769284416271257, −13.925192947954114023880878890196, −12.99086327183323970153073485521, −11.73118720968091551826244127909, −11.20413359620663629663097227810, −10.41519701414894583971718985799, −9.1827281728133724954207565983, −8.02727321008162586812390210996, −7.13368156040854333686094382334, −5.38418621080416645722564179564, −4.48390146212433485181718626682, −3.35563978747644774682909133895, −2.36068897064143373541100244518, −0.70281614558089764944720172950, 1.505858908910709867266035287711, 3.57985533770492518538028945752, 4.6049839533594278015772392589, 5.31801968723221122855280933499, 6.68346870101463012463465921942, 7.71536067525408893742604113286, 8.549132348711030899470248993213, 9.23018636942288015202564249018, 10.88892664285987735853563090209, 12.04212790535232251989863933331, 12.74418840651999564712199014853, 14.036139659094455147364576234731, 14.79529524742225068831699467991, 15.63646348456322090729692029193, 16.57713703954110388581290786073, 17.2481180094516075743041698535, 18.38451907650651225187612581544, 19.15926704858582005357924122547, 20.44580358176200251427464974714, 21.36492179287009865893482224764, 22.24643656334935931497391425503, 23.43247781038244029755205370806, 23.98598102969278470174861765992, 24.61200505808433923366764040592, 25.544687774826497924409426140630

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