L-function 1-675-675.259-r0-0-0

• Label: 1-675-675.259-r0-0-0
• Degree: $1$
• Conductor: $675$
• Sign: $0.861 - 0.508i$
• Analytic cond.: $3.13468$
• Root an. cond.: $3.13468$
• Motivic weight: $0$
• Arithmetic: yes
• Rational: no
• Primitive: yes
• Self-dual: no
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

+ (0.882 − 0.469i)2^-s + (0.559 − 0.829i)4^-s + (0.939 + 0.342i)7^-s + (0.104 − 0.994i)8^-s + (0.848 + 0.529i)11^-s + (0.882 + 0.469i)13^-s + (0.990 − 0.139i)14^-s + (−0.374 − 0.927i)16^-s + (−0.913 + 0.406i)17^-s + (−0.104 + 0.994i)19^-s + (0.997 + 0.0697i)22^-s + (0.615 + 0.788i)23^-s + 26^-s + (0.809 − 0.587i)28^-s + (−0.719 − 0.694i)29^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.861 - 0.508i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$675$    =    $3^{3} \cdot 5^{2}$
Sign:	$0.861 - 0.508i$
Analytic conductor:	$3.13468$
Root analytic conductor:	$3.13468$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{675} (259, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 675,\ (0:\ ),\ 0.861 - 0.508i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$2.762258445 - 0.7539413882i$
$L(1)$	$\approx$	$1.942709355 - 0.4533009111i$

Euler product

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

	$p$	$F_p(T)$
bad	3	$1$
	5	$1$
good	2	$1 + (0.882 - 0.469i)T$
	7	$1 + (0.939 + 0.342i)T$
	11	$1 + (0.848 + 0.529i)T$
	13	$1 + (0.882 + 0.469i)T$
	17	$1 + (-0.913 + 0.406i)T$
	19	$1 + (-0.104 + 0.994i)T$
	23	$1 + (0.615 + 0.788i)T$
	29	$1 + (-0.719 - 0.694i)T$
	31	$1 + (0.961 + 0.275i)T$

Imaginary part of the first few zeros on the critical line

−22.80469633562564193899463501269, −22.116705829125009318509226488818, −21.32150537945372011293579950078, −20.47427136514726196188267870158, −19.95523381403429783737790551730, −18.60494891010263299682113767295, −17.6343992864506466150010786875, −17.01490682651188912771405926260, −16.12988683539932587235519979825, −15.19252780521166542738747674956, −14.61145954788056638062657010327, −13.52635275618872119366597714067, −13.274539332206901243937838529881, −11.82343940920026464981123010656, −11.339577762764698579831463721900, −10.51457722270727355145555055413, −8.766298779793409026175898213449, −8.41526969025726755206392904522, −7.09462775574903430635124196305, −6.52995584968496063550570617047, −5.35747688755287101929877869563, −4.57324532498435459828866831554, −3.6840562001207333813264387787, −2.60346664584312360357850874862, −1.26902055401810138508715588531, 1.497718707919327521993289265193, 1.9823965426151065199193493717, 3.50519595143591922825080533974, 4.24845707925616089214483222506, 5.161338887566840101011281270399, 6.15700599220203293593509832501, 6.95340713550200998741505802006, 8.23831652548530561213031179680, 9.20859891450050672789265900502, 10.2302237482380787990463407271, 11.31518201381730902874236072429, 11.64384176398676948768282974419, 12.65910714815773428514825432721, 13.567619458611965780734630704336, 14.34584841600938150692290855276, 15.0601123135294273690186718340, 15.76016827006552620675420610314, 16.936101062736827055407779288779, 17.83315735890884807354259530638, 18.80331012859509516414268325707, 19.53741446081031159279068395628, 20.50072083884887232137890650534, 21.10271621385843654685324016093, 21.75348931437269995024022046202, 22.74555365519081184681203926460

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