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Properties

Label 1-475-475.141-r1-0-0
Degree $1$
Conductor $475$
Sign $-0.991 + 0.132i$
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  + (−0.669 + 0.743i)2-s + (−0.913 − 0.406i)3-s + (−0.104 − 0.994i)4-s + (0.913 − 0.406i)6-s + 7-s + (0.809 + 0.587i)8-s + (0.669 + 0.743i)9-s + (0.309 + 0.951i)11-s + (−0.309 + 0.951i)12-s + (−0.669 − 0.743i)13-s + (−0.669 + 0.743i)14-s + (−0.978 + 0.207i)16-s + (−0.104 + 0.994i)17-s − 18-s + (−0.913 − 0.406i)21-s + (−0.913 − 0.406i)22-s + ⋯
L(s)  = 1  + (−0.669 + 0.743i)2-s + (−0.913 − 0.406i)3-s + (−0.104 − 0.994i)4-s + (0.913 − 0.406i)6-s + 7-s + (0.809 + 0.587i)8-s + (0.669 + 0.743i)9-s + (0.309 + 0.951i)11-s + (−0.309 + 0.951i)12-s + (−0.669 − 0.743i)13-s + (−0.669 + 0.743i)14-s + (−0.978 + 0.207i)16-s + (−0.104 + 0.994i)17-s − 18-s + (−0.913 − 0.406i)21-s + (−0.913 − 0.406i)22-s + ⋯

Functional equation

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

Invariants

Degree: \(1\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $-0.991 + 0.132i$
Analytic conductor: \(51.0458\)
Root analytic conductor: \(51.0458\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (141, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 475,\ (1:\ ),\ -0.991 + 0.132i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.02810187116 + 0.4235867989i\)
\(L(\frac12)\) \(\approx\) \(0.02810187116 + 0.4235867989i\)
\(L(1)\) \(\approx\) \(0.5450947947 + 0.1799695147i\)
\(L(1)\) \(\approx\) \(0.5450947947 + 0.1799695147i\)

Euler product

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

−22.92009644287992145370861965937, −22.160217262231231276139298503, −21.28650023044786366734704285735, −20.972010816588852539090552817810, −19.741067342248229938311756527309, −18.8454359381781835605130139910, −17.963602846744824045043550811, −17.38031405835806540666710969438, −16.51793296472406592587547071671, −15.8588682204177279054818802049, −14.45949286779479483142254525908, −13.52405282641818117756507187649, −12.18469852390599001029224252689, −11.545976940753131528148794267116, −11.12137047796346870480755767962, −9.97298560607123046885177180287, −9.267935725323984876911875151087, −8.14941706632928053763090774201, −7.17652455400137408903853489873, −5.95522261836624168244209363533, −4.69412035668084612138526070443, −4.006872531753329338318762861405, −2.52900971342556780528267133234, −1.25934483743817576564058918517, −0.1805321839059242104897472711, 1.22634307723370385434941220574, 2.05687422709504700416067385924, 4.419984797918934082789790037174, 5.13285225877788165021479762144, 6.07942897905822553284394179627, 7.07721011781623530722392348013, 7.78843555259788228338613273409, 8.68004922088903975642978183488, 10.16531460016269030104729459354, 10.5172418071188434716644951940, 11.746812046880287942729712572, 12.50956686511650441103288259570, 13.769730171352606599428221543851, 14.7779486742874754559348333532, 15.426845553357124217471922509117, 16.54953576947954350792741938329, 17.45121942480223832629231540700, 17.68384285854885024978674565512, 18.53694256374202939219854818766, 19.600608808810146642428507941311, 20.35175298772430796800748995395, 21.72997464964639147755772361570, 22.54431521447026673505676249241, 23.43420604979916472087081280398, 24.08133244450472420485672118529

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