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Properties

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 $0$

Origins

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Normalization:  

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 + ⋯
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

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

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

\(L(\frac{1}{2})\) \(\approx\) \(0.1333210161 - 0.1569256245i\)
\(L(\frac12)\) \(\approx\) \(0.1333210161 - 0.1569256245i\)
\(L(1)\) \(\approx\) \(0.5367342957 + 0.1617882256i\)
\(L(1)\) \(\approx\) \(0.5367342957 + 0.1617882256i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
19 \( 1 \)
good2 \( 1 + (-0.0697 + 0.997i)T \)
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 \( 1 + (-0.669 + 0.743i)T \)
37 \( 1 + (-0.587 + 0.809i)T \)
41 \( 1 + (-0.961 - 0.275i)T \)
43 \( 1 + (-0.984 - 0.173i)T \)
47 \( 1 + (0.927 - 0.374i)T \)
53 \( 1 + (-0.139 + 0.990i)T \)
59 \( 1 + (-0.241 - 0.970i)T \)
61 \( 1 + (-0.719 - 0.694i)T \)
67 \( 1 + (0.529 - 0.848i)T \)
71 \( 1 + (0.882 + 0.469i)T \)
73 \( 1 + (-0.829 - 0.559i)T \)
79 \( 1 + (0.0348 + 0.999i)T \)
83 \( 1 + (-0.743 - 0.669i)T \)
89 \( 1 + (0.961 - 0.275i)T \)
97 \( 1 + (-0.529 - 0.848i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

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