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Properties

Label 1-475-475.142-r1-0-0
Degree $1$
Conductor $475$
Sign $0.878 + 0.477i$
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  + (0.829 + 0.559i)2-s + (−0.469 + 0.882i)3-s + (0.374 + 0.927i)4-s + (−0.882 + 0.469i)6-s + (0.866 − 0.5i)7-s + (−0.207 + 0.978i)8-s + (−0.559 − 0.829i)9-s + (−0.104 − 0.994i)11-s + (−0.994 − 0.104i)12-s + (−0.898 − 0.438i)13-s + (0.997 + 0.0697i)14-s + (−0.719 + 0.694i)16-s + (0.788 − 0.615i)17-s − i·18-s + (0.0348 + 0.999i)21-s + (0.469 − 0.882i)22-s + ⋯
L(s)  = 1  + (0.829 + 0.559i)2-s + (−0.469 + 0.882i)3-s + (0.374 + 0.927i)4-s + (−0.882 + 0.469i)6-s + (0.866 − 0.5i)7-s + (−0.207 + 0.978i)8-s + (−0.559 − 0.829i)9-s + (−0.104 − 0.994i)11-s + (−0.994 − 0.104i)12-s + (−0.898 − 0.438i)13-s + (0.997 + 0.0697i)14-s + (−0.719 + 0.694i)16-s + (0.788 − 0.615i)17-s − i·18-s + (0.0348 + 0.999i)21-s + (0.469 − 0.882i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.878 + 0.477i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.878 + 0.477i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $0.878 + 0.477i$
Analytic conductor: \(51.0458\)
Root analytic conductor: \(51.0458\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (142, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 475,\ (1:\ ),\ 0.878 + 0.477i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.733323700 + 0.6943178670i\)
\(L(\frac12)\) \(\approx\) \(2.733323700 + 0.6943178670i\)
\(L(1)\) \(\approx\) \(1.443721658 + 0.6111377900i\)
\(L(1)\) \(\approx\) \(1.443721658 + 0.6111377900i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
19 \( 1 \)
good2 \( 1 + (0.829 + 0.559i)T \)
3 \( 1 + (-0.469 + 0.882i)T \)
7 \( 1 + (0.866 - 0.5i)T \)
11 \( 1 + (-0.104 - 0.994i)T \)
13 \( 1 + (-0.898 - 0.438i)T \)
17 \( 1 + (0.788 - 0.615i)T \)
23 \( 1 + (0.970 + 0.241i)T \)
29 \( 1 + (0.615 - 0.788i)T \)
31 \( 1 + (0.669 - 0.743i)T \)
37 \( 1 + (0.587 - 0.809i)T \)
41 \( 1 + (-0.719 + 0.694i)T \)
43 \( 1 + (0.642 - 0.766i)T \)
47 \( 1 + (-0.788 - 0.615i)T \)
53 \( 1 + (-0.927 + 0.374i)T \)
59 \( 1 + (-0.961 - 0.275i)T \)
61 \( 1 + (-0.241 + 0.970i)T \)
67 \( 1 + (0.999 + 0.0348i)T \)
71 \( 1 + (0.848 - 0.529i)T \)
73 \( 1 + (0.898 - 0.438i)T \)
79 \( 1 + (0.882 + 0.469i)T \)
83 \( 1 + (-0.743 - 0.669i)T \)
89 \( 1 + (0.719 + 0.694i)T \)
97 \( 1 + (-0.999 + 0.0348i)T \)
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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

−23.48809353197675298169418587577, −22.80088689180503986076177130885, −21.85613505768553201444311060131, −21.16328062093924014595006499812, −20.161438051334798626961187818201, −19.29125624028555252211943715636, −18.58961661606443595522463244212, −17.67543842452935994029110471920, −16.84231898511186616558269613971, −15.48609281665743807981446125938, −14.57600182268191328379462777114, −14.046414923604405460036391561502, −12.64889687955464456504694747742, −12.41126457343096163959248213621, −11.517166412096028345226623156546, −10.67604492282026724677881032157, −9.61276358346246249878331970342, −8.20485401746315588816483827856, −7.159067275906476705101165779767, −6.31775389132230298146584532722, −5.07477612839484611226122256024, −4.72975071395293022685234904988, −2.95379003768309818479826786422, −1.97542914146955411994133935575, −1.1731057232674284754546099159, 0.63445460213077311013472728302, 2.70544453852108027390499870291, 3.66465880053993323068473106140, 4.74975517344491889264443261104, 5.28856747210675980342689230061, 6.2831518913770976558204328689, 7.515683535368158091764507787, 8.30953432368038808906096014229, 9.56187412057002407399626293991, 10.7304998786472921572058887338, 11.45663869745790694399608591425, 12.21425705326480506673470665510, 13.500683406314545186771805036925, 14.296997816861843020952990849596, 15.0275155283168257779243176227, 15.82719441474119174221315802433, 16.8861243435381882970146509272, 17.120574409351063848908981424608, 18.242941517656029104540501258924, 19.722175240689603726935623342284, 20.79123410735815186917574947583, 21.220145601266397641513596945211, 22.00170705440761152607250541951, 22.91513017392935436186688406480, 23.49240093569704689789630294236

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