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Properties

Label 1-475-475.386-r0-0-0
Degree $1$
Conductor $475$
Sign $0.791 - 0.611i$
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  + (−0.882 − 0.469i)2-s + (−0.241 − 0.970i)3-s + (0.559 + 0.829i)4-s + (−0.241 + 0.970i)6-s + (−0.5 − 0.866i)7-s + (−0.104 − 0.994i)8-s + (−0.882 + 0.469i)9-s + (0.669 + 0.743i)11-s + (0.669 − 0.743i)12-s + (0.848 + 0.529i)13-s + (0.0348 + 0.999i)14-s + (−0.374 + 0.927i)16-s + (0.438 + 0.898i)17-s + 18-s + (−0.719 + 0.694i)21-s + (−0.241 − 0.970i)22-s + ⋯
L(s)  = 1  + (−0.882 − 0.469i)2-s + (−0.241 − 0.970i)3-s + (0.559 + 0.829i)4-s + (−0.241 + 0.970i)6-s + (−0.5 − 0.866i)7-s + (−0.104 − 0.994i)8-s + (−0.882 + 0.469i)9-s + (0.669 + 0.743i)11-s + (0.669 − 0.743i)12-s + (0.848 + 0.529i)13-s + (0.0348 + 0.999i)14-s + (−0.374 + 0.927i)16-s + (0.438 + 0.898i)17-s + 18-s + (−0.719 + 0.694i)21-s + (−0.241 − 0.970i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.791 - 0.611i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.791 - 0.611i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $0.791 - 0.611i$
Analytic conductor: \(2.20589\)
Root analytic conductor: \(2.20589\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (386, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 475,\ (0:\ ),\ 0.791 - 0.611i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7157692835 - 0.2441431896i\)
\(L(\frac12)\) \(\approx\) \(0.7157692835 - 0.2441431896i\)
\(L(1)\) \(\approx\) \(0.6338380907 - 0.2450370873i\)
\(L(1)\) \(\approx\) \(0.6338380907 - 0.2450370873i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
19 \( 1 \)
good2 \( 1 + (-0.882 - 0.469i)T \)
3 \( 1 + (-0.241 - 0.970i)T \)
7 \( 1 + (-0.5 - 0.866i)T \)
11 \( 1 + (0.669 + 0.743i)T \)
13 \( 1 + (0.848 + 0.529i)T \)
17 \( 1 + (0.438 + 0.898i)T \)
23 \( 1 + (-0.615 + 0.788i)T \)
29 \( 1 + (0.438 - 0.898i)T \)
31 \( 1 + (0.913 + 0.406i)T \)
37 \( 1 + (0.309 + 0.951i)T \)
41 \( 1 + (-0.374 + 0.927i)T \)
43 \( 1 + (-0.939 + 0.342i)T \)
47 \( 1 + (0.438 - 0.898i)T \)
53 \( 1 + (0.559 + 0.829i)T \)
59 \( 1 + (0.990 - 0.139i)T \)
61 \( 1 + (-0.615 + 0.788i)T \)
67 \( 1 + (-0.719 - 0.694i)T \)
71 \( 1 + (0.961 + 0.275i)T \)
73 \( 1 + (0.848 - 0.529i)T \)
79 \( 1 + (-0.241 - 0.970i)T \)
83 \( 1 + (0.913 + 0.406i)T \)
89 \( 1 + (-0.374 - 0.927i)T \)
97 \( 1 + (-0.719 + 0.694i)T \)
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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

−24.06616214761305694680812980699, −22.935300382944777711063903219713, −22.32864487883728109499438028345, −21.27612976393927351965400922499, −20.45784269965652157472513289642, −19.58681506577609489166290404947, −18.619839441830172581873997031397, −17.931891914451701937859689489089, −16.84266615620967454605272831814, −16.11867650368749461366477685343, −15.68329030203770655037225650521, −14.68818558024680112977255746531, −13.868441967075862496752685273560, −12.20248144654314449237955261437, −11.40115155935209994250497207566, −10.5147604096784674514520953428, −9.64968095627150288569744845878, −8.85648460070618543210328439747, −8.25400918268463202806244238933, −6.65488100749771937206165696417, −5.91491636743125775169946072272, −5.15146362171028133202158720821, −3.61551437517273949029105733278, −2.56041458069383117112079541887, −0.72896928535148704914267448630, 1.06986494675712147929010028228, 1.81499826782700755363762755757, 3.22732375010394841497695448817, 4.24923020388649948900971827666, 6.20589949954265199683550365367, 6.736766454275804538216335781946, 7.72157351971172116764514900209, 8.4945678269971287416673945001, 9.698490068578808063828511249018, 10.4617040684922512519467835341, 11.581860112247889593641408973537, 12.12734074930239662077942510556, 13.2021518258758436494249470756, 13.807705767848605261194826158286, 15.24203901446399084757751335486, 16.50567748458776038704051934970, 17.03358981157438208363684189067, 17.796095652370263540523801938719, 18.639010411967987777419514096473, 19.555843098085785928630546998686, 19.88965321412767332671174915236, 20.94977056067293226688721089622, 21.99666712091782807085809568198, 23.05981877192504858113895360331, 23.63821986064845028158716308035

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