L-function 1-475-475.167-r0-0-0

• Label: 1-475-475.167-r0-0-0
• Degree: $1$
• Conductor: $475$
• Sign: $-0.161 - 0.986i$
• 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

+ (−0.0697 + 0.997i)2^-s + (−0.999 + 0.0348i)3^-s + (−0.990 − 0.139i)4^-s + (0.0348 − 0.999i)6^-s + (0.866 − 0.5i)7^-s + (0.207 − 0.978i)8^-s + (0.997 − 0.0697i)9^-s + (−0.104 − 0.994i)11^-s + (0.994 + 0.104i)12^-s + (−0.829 + 0.559i)13^-s + (0.438 + 0.898i)14^-s + (0.961 + 0.275i)16^-s + (−0.927 − 0.374i)17^-s + i·18^-s + (−0.848 + 0.529i)21^-s + (0.999 − 0.0348i)22^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.161 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$475$    =    $5^{2} \cdot 19$
Sign:	$-0.161 - 0.986i$
Analytic conductor:	$2.20589$
Root analytic conductor:	$2.20589$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{475} (167, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 475,\ (0:\ ),\ -0.161 - 0.986i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$0.1333210161 - 0.1569256245i$
$L(1)$	$\approx$	$0.5367342957 + 0.1617882256i$

Euler product

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

	$p$	$F_p(T)$
bad	5	$1$
	19	$1$
good	2	$1 + (-0.0697 + 0.997i)T$
	3	$1 + (-0.999 + 0.0348i)T$
	7	$1 + (0.866 - 0.5i)T$
	11	$1 + (-0.104 - 0.994i)T$
	13	$1 + (-0.829 + 0.559i)T$
	17	$1 + (-0.927 - 0.374i)T$
	23	$1 + (-0.694 + 0.719i)T$
	29	$1 + (-0.374 - 0.927i)T$
	31	$1 + (-0.669 + 0.743i)T$

Imaginary part of the first few zeros on the critical line

−23.9219421539163958787133950309, −22.953141597981341278744071557198, −22.13387766945670501483264085395, −21.75101009853809565455116391577, −20.61032610394405936842791177099, −19.99461337528149176882811064324, −18.78139902021523314713649718898, −17.95163843388977893092972989972, −17.62289283838434021051922336852, −16.66489661170362045186776239020, −15.27687598539959790433966627329, −14.58883654491750604595661038741, −13.2013379018304523701748423908, −12.42840858217178443056354964714, −11.86886566370897856282399631840, −10.89833897591511087351092839645, −10.26390387944850046426730544265, −9.2625190772717535116995975034, −8.11483636693134938531970646274, −7.076571570482413378909038449, −5.60096385159983213781283412215, −4.874461881643195052999566976636, −4.08791945503839655182143585869, −2.374910207679496170119374290, −1.60334888863503428116304653724, 0.1346123206332826723647766719, 1.63384810274201224227647423388, 3.79901226724437941443107295606, 4.76170621732006360441525006550, 5.42979356358868513102909922874, 6.51543110967549587284381417620, 7.27945438982369828497844691289, 8.21458925140201225151474268643, 9.3430394058808024571672388909, 10.36208632036673481289814623298, 11.28554905803469003166198349475, 12.15319001457048032052522573635, 13.48125284958253264489073329495, 13.98354350375180449119483566736, 15.19475608530348766138949353401, 15.91957481820657034078495515879, 16.95566593531093496897989922886, 17.231724953932246318317000913374, 18.23924422361244577905102932265, 18.89884879624218944540147214371, 20.15125896358042915984658952256, 21.63735098231433108874630069325, 21.84879959907493109814035824471, 22.97082264342892682540669957214, 23.84768004508762379737842677829

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