Properties

Label 1-475-475.56-r1-0-0
Degree $1$
Conductor $475$
Sign $0.535 + 0.844i$
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  + (0.809 − 0.587i)2-s + (−0.309 + 0.951i)3-s + (0.309 − 0.951i)4-s + (0.309 + 0.951i)6-s + 7-s + (−0.309 − 0.951i)8-s + (−0.809 − 0.587i)9-s + (−0.809 + 0.587i)11-s + (0.809 + 0.587i)12-s + (0.809 + 0.587i)13-s + (0.809 − 0.587i)14-s + (−0.809 − 0.587i)16-s + (0.309 + 0.951i)17-s − 18-s + (−0.309 + 0.951i)21-s + (−0.309 + 0.951i)22-s + ⋯
L(s)  = 1  + (0.809 − 0.587i)2-s + (−0.309 + 0.951i)3-s + (0.309 − 0.951i)4-s + (0.309 + 0.951i)6-s + 7-s + (−0.309 − 0.951i)8-s + (−0.809 − 0.587i)9-s + (−0.809 + 0.587i)11-s + (0.809 + 0.587i)12-s + (0.809 + 0.587i)13-s + (0.809 − 0.587i)14-s + (−0.809 − 0.587i)16-s + (0.309 + 0.951i)17-s − 18-s + (−0.309 + 0.951i)21-s + (−0.309 + 0.951i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.535 + 0.844i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.535 + 0.844i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $0.535 + 0.844i$
Analytic conductor: \(51.0458\)
Root analytic conductor: \(51.0458\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (56, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 475,\ (1:\ ),\ 0.535 + 0.844i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.382213936 + 1.309633194i\)
\(L(\frac12)\) \(\approx\) \(2.382213936 + 1.309633194i\)
\(L(1)\) \(\approx\) \(1.562000044 + 0.09827271304i\)
\(L(1)\) \(\approx\) \(1.562000044 + 0.09827271304i\)

Euler product

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

−23.541582181242197705123391603150, −22.914967387428380683905915325, −22.00472401570690172222443546889, −20.8937368838309154616930058920, −20.44463912300733813775173655542, −19.0200662933830947429375151200, −17.96615830627296101776971923425, −17.714850197677633261783208729585, −16.41279684285965479360016516011, −15.819555792731951423928354121648, −14.525600342738477975376296537064, −13.91951833619876493298109517406, −13.1423025573747630526208525881, −12.2940145835368847624685999509, −11.37914831592754625644559081475, −10.74216221426197827448896785560, −8.75103692445167290454972409727, −7.90835444860019480281094491416, −7.43334992311980320399932119960, −6.074271361435874899630330210026, −5.55127283695531284188922896392, −4.52231249272359272227519974011, −3.083892463198982602325693371800, −2.08456493679202601962358998546, −0.564732304760017636561915565025, 1.27756983396693740410237665064, 2.44798078502029631672818545814, 3.8170735351380742220187626507, 4.412500304871735401232179295444, 5.41755850142900032064871541644, 6.10675003724037026459703018275, 7.64264514053143736299322519795, 8.89154600098289216497632834487, 9.94461400952548313513926780616, 10.75234907138808571228911321415, 11.35950727961730788569483227131, 12.23767742323704531366387285177, 13.3227925450226960415301406648, 14.34721275498256305434832486758, 14.99613766105090790036820166775, 15.73792507864737970143387741057, 16.69675302168634205586181991861, 17.86293153732406559315820821720, 18.61082518221396346470479981159, 19.96401356488614128527221454378, 20.58678047101553881510829309908, 21.33583120546365815600848685068, 21.7697999069437992724396080009, 22.85130206193749783912156299147, 23.65765073494971490069889177682

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