L-function 1-297-297.124-r0-0-0

• Label: 1-297-297.124-r0-0-0
• Degree: $1$
• Conductor: $297$
• Sign: $0.320 + 0.947i$
• 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

+ (0.990 + 0.139i)2^-s + (0.961 + 0.275i)4^-s + (−0.374 + 0.927i)5^-s + (0.559 + 0.829i)7^-s + (0.913 + 0.406i)8^-s + (−0.5 + 0.866i)10^-s + (−0.882 − 0.469i)13^-s + (0.438 + 0.898i)14^-s + (0.848 + 0.529i)16^-s + (−0.978 − 0.207i)17^-s + (0.913 + 0.406i)19^-s + (−0.615 + 0.788i)20^-s + (−0.939 + 0.342i)23^-s + (−0.719 − 0.694i)25^-s + (−0.809 − 0.587i)26^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 297 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.320 + 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$297$    =    $3^{3} \cdot 11$
Sign:	$0.320 + 0.947i$
Analytic conductor:	$1.37926$
Root analytic conductor:	$1.37926$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{297} (124, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 297,\ (0:\ ),\ 0.320 + 0.947i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$1.800641718 + 1.291618702i$
$L(1)$	$\approx$	$1.694144000 + 0.6380581422i$

Euler product

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

	$p$	$F_p(T)$
bad	3	$1$
	11	$1$
good	2	$1 + (0.990 + 0.139i)T$
	5	$1 + (-0.374 + 0.927i)T$
	7	$1 + (0.559 + 0.829i)T$
	13	$1 + (-0.882 - 0.469i)T$
	17	$1 + (-0.978 - 0.207i)T$
	19	$1 + (0.913 + 0.406i)T$
	23	$1 + (-0.939 + 0.342i)T$
	29	$1 + (0.438 - 0.898i)T$
	31	$1 + (0.0348 + 0.999i)T$

Imaginary part of the first few zeros on the critical line

−24.71807402232376732123523252968, −24.11972106191400555868349012844, −23.70467327452170671393868187447, −22.48263582581793503932309070052, −21.71766562573543317592342910378, −20.5995923980609907883997153908, −20.117915674091439017692262491334, −19.37714801246734289330447996816, −17.76376832497049180085007080664, −16.75824765830153345988296233270, −16.04194222751267453796077527200, −15.00370327064771597251043682679, −14.01334054678914868004984418635, −13.273142059684876357683203821775, −12.23458707034068667435220705355, −11.51261826652725842495605535174, −10.49657149597928022410130589626, −9.24471306559886262685185991391, −7.842942678134959058458421952654, −7.07872682548862942673567448100, −5.68757715107887982602669445241, −4.52482845187939871949186826279, −4.154781823949846642273703600889, −2.4847560563272372061199803112, −1.14966247100462094236253568756, 2.13017803168992016729466504894, 2.9296894070621187976271198521, 4.186310001760543752903279442370, 5.29437258469125370515202753617, 6.2661130458613775914124902716, 7.37698258367919008096047690307, 8.149985390819100194610184556441, 9.80841774261703048399794979620, 10.98889826683631986721512757750, 11.75012868169819734129713940116, 12.4838302562975690720758530866, 13.85163055902287632452101086324, 14.52009298493945639294363018109, 15.400931958165234430192903675028, 15.95080272391547894813145254806, 17.47261144509647935936550082083, 18.254700384200941299988859310217, 19.48102791527431384023622720471, 20.222292250305792287617694861251, 21.4601234638574454136262592507, 22.08862136586879537429303220723, 22.72596902186066527129423092434, 23.747018026375035655569697008945, 24.65238851276116195162031048099, 25.2418910879304957619302939494

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