L-function 1-475-475.238-r0-0-0

• Label: 1-475-475.238-r0-0-0
• Degree: $1$
• Conductor: $475$
• Sign: $-0.667 - 0.744i$
• Analytic cond.: $2.20589$
• Root an. cond.: $2.20589$
• Motivic weight: $0$
• Arithmetic: yes
• Rational: no
• Primitive: yes
• Self-dual: no
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

+ (0.788 − 0.615i)2^-s + (0.898 − 0.438i)3^-s + (0.241 − 0.970i)4^-s + (0.438 − 0.898i)6^-s + (−0.866 + 0.5i)7^-s + (−0.406 − 0.913i)8^-s + (0.615 − 0.788i)9^-s + (−0.978 − 0.207i)11^-s + (−0.207 − 0.978i)12^-s + (0.139 − 0.990i)13^-s + (−0.374 + 0.927i)14^-s + (−0.882 − 0.469i)16^-s + (0.275 − 0.961i)17^-s − i·18^-s + (−0.559 + 0.829i)21^-s + (−0.898 + 0.438i)22^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.667 - 0.744i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$475$    =    $5^{2} \cdot 19$
Sign:	$-0.667 - 0.744i$
Analytic conductor:	$2.20589$
Root analytic conductor:	$2.20589$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{475} (238, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 475,\ (0:\ ),\ -0.667 - 0.744i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$0.9552511354 - 2.139125898i$
$L(1)$	$\approx$	$1.408564187 - 1.153610276i$

Euler product

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

	$p$	$F_p(T)$
bad	5	$1$
	19	$1$
good	2	$1 + (0.788 - 0.615i)T$
	3	$1 + (0.898 - 0.438i)T$
	7	$1 + (-0.866 + 0.5i)T$
	11	$1 + (-0.978 - 0.207i)T$
	13	$1 + (0.139 - 0.990i)T$
	17	$1 + (0.275 - 0.961i)T$
	23	$1 + (-0.529 + 0.848i)T$
	29	$1 + (0.961 - 0.275i)T$
	31	$1 + (0.104 - 0.994i)T$

Imaginary part of the first few zeros on the critical line

−23.96994612093315929547746457408, −23.44346666211174374579348302332, −22.41783206196092408222440943492, −21.5409268372008627832460197086, −20.97694877798481524019459171653, −20.036588006148183073030440628576, −19.245192820060040022469566378366, −18.12435306325665958434399930213, −16.817554079770591079650342385592, −16.14063963548077360188227305651, −15.56795198378373184628698492551, −14.486149671329882812517069416709, −13.91694134207793062592745676541, −13.00465200809268482134232672652, −12.38217737439959796386638365121, −10.8346578869382315955570489139, −10.00236460206257164104743833164, −8.842022421668283445519456992626, −8.01408776581529370270411301339, −7.07178461427277806313862283359, −6.16674299621347577957026895890, −4.82437534827777270891878818135, −4.02302122924991095011328779437, −3.146829177021933454948988129401, −2.13704927791420243937756321020, 0.88188812455066764008865637988, 2.544063473394946269710332189569, 2.86458324508433351561236114818, 3.954156608329161142978925403309, 5.347705002017378062781632333008, 6.17799465384430132418152917464, 7.34393275513095164823919878082, 8.366622125976650549608459825814, 9.64812536132010904417359872530, 10.07827177345176928812059906503, 11.4876590308258395117443963935, 12.3845581902267780701210060474, 13.251119004138264027145673466860, 13.5549655703522959476387370496, 14.79627120658963239950746044311, 15.50256140116586051883792221678, 16.16295075809533143609451077727, 18.115179237155745182830024535801, 18.49999294231930086228030437363, 19.55164733550181180814262629112, 20.04169591172953421605328718534, 20.989563329355732963908989432519, 21.62946755360016921203743062414, 22.75483434126327033769916767808, 23.343835402229382265498075914346

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