L-function 1-475-475.196-r0-0-0

• Label: 1-475-475.196-r0-0-0
• Degree: $1$
• Conductor: $475$
• Sign: $-0.748 - 0.663i$
• 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

+ (−0.719 + 0.694i)2^-s + (−0.374 + 0.927i)3^-s + (0.0348 − 0.999i)4^-s + (−0.374 − 0.927i)6^-s + (−0.5 − 0.866i)7^-s + (0.669 + 0.743i)8^-s + (−0.719 − 0.694i)9^-s + (0.913 − 0.406i)11^-s + (0.913 + 0.406i)12^-s + (−0.241 + 0.970i)13^-s + (0.961 + 0.275i)14^-s + (−0.997 − 0.0697i)16^-s + (−0.882 − 0.469i)17^-s + 18^-s + (0.990 − 0.139i)21^-s + (−0.374 + 0.927i)22^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.748 - 0.663i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$475$    =    $5^{2} \cdot 19$
Sign:	$-0.748 - 0.663i$
Analytic conductor:	$2.20589$
Root analytic conductor:	$2.20589$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{475} (196, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 475,\ (0:\ ),\ -0.748 - 0.663i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$-0.05253337749 + 0.1385434168i$
$L(1)$	$\approx$	$0.4313615448 + 0.2404601602i$

Euler product

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

	$p$	$F_p(T)$
bad	5	$1$
	19	$1$
good	2	$1 + (-0.719 + 0.694i)T$
	3	$1 + (-0.374 + 0.927i)T$
	7	$1 + (-0.5 - 0.866i)T$
	11	$1 + (0.913 - 0.406i)T$
	13	$1 + (-0.241 + 0.970i)T$
	17	$1 + (-0.882 - 0.469i)T$
	23	$1 + (0.559 + 0.829i)T$
	29	$1 + (-0.882 + 0.469i)T$
	31	$1 + (-0.978 + 0.207i)T$

Imaginary part of the first few zeros on the critical line

−23.02639780507621663590190077514, −22.31571189028457286039913334007, −21.81737197361860149161748653190, −20.325522209295528886645945736656, −19.82729224801404812703448520850, −18.9012970103946024446034442442, −18.36299185584991089844940321570, −17.40002407287870745384281345058, −16.91028249187050615497791339901, −15.70510375332290552026568538256, −14.627929756454048465485894803305, −13.11909842961613971592312353481, −12.75901863495594029312712255725, −11.89160465161031885732242272297, −11.14072508542218159627440048526, −10.08275227742798321887978756880, −8.99172035752726310820232725992, −8.32464034518867977667764239458, −7.151687376860975905943411935497, −6.42129417504332205416141301925, −5.18372097952543026436181452236, −3.63439367932832187606482661607, −2.46904165148702125198357899699, −1.66371633453568931943032447992, −0.1093681537676066674508724041, 1.50174552266659996073792961980, 3.42006723370896710948496074773, 4.424099110634771590235667010418, 5.40476148479766796023590292347, 6.61953294644362502085627579857, 7.0592214572885186096590449523, 8.64550329849904020680834625437, 9.32928575609214847452241507314, 10.003672040414086624391314226531, 11.078856589456583191938854494460, 11.60490200286928457590399062003, 13.36161806082102307096601790501, 14.249066108110095739646738823989, 15.02198139737859871986087153008, 16.041333305891176722872326492, 16.6651370657137804092603848963, 17.13951956776382978534038267398, 18.127247421048697956258432553285, 19.34168958916650879883929796640, 19.89927858260795205501044704615, 20.82928895840679153030550775588, 22.0507016735309753778527589099, 22.641737988903709613137662229952, 23.63306317819484543009187866319, 24.241808646473000915070855389833

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