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Properties

Label 1-475-475.388-r0-0-0
Degree $1$
Conductor $475$
Sign $0.991 - 0.132i$
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  + (0.743 + 0.669i)2-s + (0.406 − 0.913i)3-s + (0.104 + 0.994i)4-s + (0.913 − 0.406i)6-s − i·7-s + (−0.587 + 0.809i)8-s + (−0.669 − 0.743i)9-s + (0.309 + 0.951i)11-s + (0.951 + 0.309i)12-s + (0.743 − 0.669i)13-s + (0.669 − 0.743i)14-s + (−0.978 + 0.207i)16-s + (0.994 + 0.104i)17-s − i·18-s + (−0.913 − 0.406i)21-s + (−0.406 + 0.913i)22-s + ⋯
L(s)  = 1  + (0.743 + 0.669i)2-s + (0.406 − 0.913i)3-s + (0.104 + 0.994i)4-s + (0.913 − 0.406i)6-s − i·7-s + (−0.587 + 0.809i)8-s + (−0.669 − 0.743i)9-s + (0.309 + 0.951i)11-s + (0.951 + 0.309i)12-s + (0.743 − 0.669i)13-s + (0.669 − 0.743i)14-s + (−0.978 + 0.207i)16-s + (0.994 + 0.104i)17-s − i·18-s + (−0.913 − 0.406i)21-s + (−0.406 + 0.913i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.132i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.132i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $0.991 - 0.132i$
Analytic conductor: \(2.20589\)
Root analytic conductor: \(2.20589\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (388, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 475,\ (0:\ ),\ 0.991 - 0.132i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.339206074 - 0.1551891321i\)
\(L(\frac12)\) \(\approx\) \(2.339206074 - 0.1551891321i\)
\(L(1)\) \(\approx\) \(1.772593098 + 0.05372375450i\)
\(L(1)\) \(\approx\) \(1.772593098 + 0.05372375450i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
19 \( 1 \)
good2 \( 1 + (0.743 + 0.669i)T \)
3 \( 1 + (0.406 - 0.913i)T \)
7 \( 1 - iT \)
11 \( 1 + (0.309 + 0.951i)T \)
13 \( 1 + (0.743 - 0.669i)T \)
17 \( 1 + (0.994 + 0.104i)T \)
23 \( 1 + (0.207 - 0.978i)T \)
29 \( 1 + (-0.104 - 0.994i)T \)
31 \( 1 + (0.809 + 0.587i)T \)
37 \( 1 + (0.951 + 0.309i)T \)
41 \( 1 + (0.978 - 0.207i)T \)
43 \( 1 + (0.866 - 0.5i)T \)
47 \( 1 + (-0.994 + 0.104i)T \)
53 \( 1 + (-0.994 + 0.104i)T \)
59 \( 1 + (-0.978 + 0.207i)T \)
61 \( 1 + (-0.978 - 0.207i)T \)
67 \( 1 + (0.406 + 0.913i)T \)
71 \( 1 + (-0.913 - 0.406i)T \)
73 \( 1 + (0.743 + 0.669i)T \)
79 \( 1 + (0.913 + 0.406i)T \)
83 \( 1 + (0.587 - 0.809i)T \)
89 \( 1 + (-0.978 - 0.207i)T \)
97 \( 1 + (-0.406 + 0.913i)T \)
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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

−23.65442168451692099504211005646, −22.7323559867797662210737529267, −21.87928900711958637460834182935, −21.29843637562013638390992029176, −20.85910587792725335287780667201, −19.59327916545431304566091568418, −19.09545787549896416434233546022, −18.17783302206020466516968800425, −16.56204885064689468707697580100, −15.92505839954516506787027087958, −15.047959585289966628665986520501, −14.266845507720298855341079829378, −13.595721100260562313774370798569, −12.43034354443041850562151335468, −11.38814691498248363095383540039, −10.96561601229268997259890237436, −9.58280420031914734288949008209, −9.17075847085678772671777350225, −8.0298672130377535342797483678, −6.20439436746547490511575546975, −5.58008507037869334546045348920, −4.536380944953793000756663215672, −3.46819479339150292376030723178, −2.84082479535406066878218274550, −1.49344499670233260224339816171, 1.13567676783902985109852982253, 2.616469015190935387110793308579, 3.6387423951601230203459934659, 4.60568326268322299733050112739, 5.98479516042201694435800427566, 6.702982174995339026153127589851, 7.66970857821016794327105531938, 8.134501859962373195006873186648, 9.4509747092350806402726721277, 10.77279726606299666956797242659, 11.98570126686968990715135282942, 12.71042770388345843095288024843, 13.43174951967008667949686152916, 14.25669753811558625273976803027, 14.87250314394820251158217878694, 15.96219648371471833314529783664, 17.06678059468943891939541499644, 17.580524172131932459266125070683, 18.54819768476289604445425215498, 19.72795280703042182054513771322, 20.540587278538236409442709609364, 21.05212826892307195158820714818, 22.72755615677912275712556448956, 22.965997011924011128676398667555, 23.73825889070248184741249724776

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