Properties

Label 1-475-475.356-r1-0-0
Degree $1$
Conductor $475$
Sign $0.566 - 0.824i$
Analytic cond. $51.0458$
Root an. cond. $51.0458$
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  + (0.241 − 0.970i)2-s + (0.615 − 0.788i)3-s + (−0.882 − 0.469i)4-s + (−0.615 − 0.788i)6-s + (−0.5 + 0.866i)7-s + (−0.669 + 0.743i)8-s + (−0.241 − 0.970i)9-s + (0.913 + 0.406i)11-s + (−0.913 + 0.406i)12-s + (−0.961 − 0.275i)13-s + (0.719 + 0.694i)14-s + (0.559 + 0.829i)16-s + (0.848 + 0.529i)17-s − 18-s + (0.374 + 0.927i)21-s + (0.615 − 0.788i)22-s + ⋯
L(s)  = 1  + (0.241 − 0.970i)2-s + (0.615 − 0.788i)3-s + (−0.882 − 0.469i)4-s + (−0.615 − 0.788i)6-s + (−0.5 + 0.866i)7-s + (−0.669 + 0.743i)8-s + (−0.241 − 0.970i)9-s + (0.913 + 0.406i)11-s + (−0.913 + 0.406i)12-s + (−0.961 − 0.275i)13-s + (0.719 + 0.694i)14-s + (0.559 + 0.829i)16-s + (0.848 + 0.529i)17-s − 18-s + (0.374 + 0.927i)21-s + (0.615 − 0.788i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.566 - 0.824i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.566 - 0.824i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $0.566 - 0.824i$
Analytic conductor: \(51.0458\)
Root analytic conductor: \(51.0458\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (356, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 475,\ (1:\ ),\ 0.566 - 0.824i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.968189485 - 1.035540621i\)
\(L(\frac12)\) \(\approx\) \(1.968189485 - 1.035540621i\)
\(L(1)\) \(\approx\) \(1.096814629 - 0.6953335436i\)
\(L(1)\) \(\approx\) \(1.096814629 - 0.6953335436i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
19 \( 1 \)
good2 \( 1 + (0.241 - 0.970i)T \)
3 \( 1 + (0.615 - 0.788i)T \)
7 \( 1 + (-0.5 + 0.866i)T \)
11 \( 1 + (0.913 + 0.406i)T \)
13 \( 1 + (-0.961 - 0.275i)T \)
17 \( 1 + (0.848 + 0.529i)T \)
23 \( 1 + (0.438 + 0.898i)T \)
29 \( 1 + (-0.848 + 0.529i)T \)
31 \( 1 + (0.978 + 0.207i)T \)
37 \( 1 + (0.809 + 0.587i)T \)
41 \( 1 + (-0.559 - 0.829i)T \)
43 \( 1 + (0.173 + 0.984i)T \)
47 \( 1 + (0.848 - 0.529i)T \)
53 \( 1 + (0.882 + 0.469i)T \)
59 \( 1 + (0.997 - 0.0697i)T \)
61 \( 1 + (0.438 + 0.898i)T \)
67 \( 1 + (0.374 - 0.927i)T \)
71 \( 1 + (-0.990 - 0.139i)T \)
73 \( 1 + (0.961 - 0.275i)T \)
79 \( 1 + (0.615 - 0.788i)T \)
83 \( 1 + (-0.978 - 0.207i)T \)
89 \( 1 + (-0.559 + 0.829i)T \)
97 \( 1 + (0.374 + 0.927i)T \)
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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

−23.82117822871398923921835541722, −22.70495396068069363311628534970, −22.311373637466900795824547794552, −21.33343865380841874677665485270, −20.44611950605549626962917949502, −19.43792273664154510238500368684, −18.74740972802602334602167311252, −17.143516881430302035369815316735, −16.789210413008971597363236215179, −16.072698113913606331507818037689, −14.99065373454633408143814980544, −14.32213800185852521471562283524, −13.73626907606431871183213714324, −12.71193848620783173009783706749, −11.483240145029992988582803476047, −10.05680248577401651529484458587, −9.55141816469414917683066199018, −8.57894675405669812442840096819, −7.56662853646179098534447483439, −6.77794485556097202314799306555, −5.5654943799271568272111070110, −4.456655714083927781706195584494, −3.8120189978491800276752197420, −2.7553934041134106476976986422, −0.60244506285808636745005588617, 0.99329668779134148713908495540, 2.04029916443044728931004813113, 2.949230952740299721890512265263, 3.81947706261358377901134852487, 5.26035613849506677542693936113, 6.23398732251900491182163099098, 7.42746396689653911131444505427, 8.59415110532965869516990214044, 9.37602142976270947475710043535, 10.054959104838749903834683435, 11.611894963251327479708905648746, 12.19653317757780723251034004679, 12.81367507053126751436568140226, 13.73474171435347608512550193652, 14.77975822209492849966009854879, 15.13008298214807502187504456441, 16.9359268068046432405429803645, 17.77593588295319843203598593771, 18.70451110653983860615065933051, 19.37121564711594391411821962056, 19.8400736806172645836010816899, 20.84617806995610181953115113131, 21.76021832224481114602167220784, 22.498116009718116568633603667664, 23.3682685577051536987630453778

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