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Properties

Label 1-475-475.446-r0-0-0
Degree $1$
Conductor $475$
Sign $0.998 + 0.0576i$
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  + (0.961 + 0.275i)2-s + (0.990 − 0.139i)3-s + (0.848 + 0.529i)4-s + (0.990 + 0.139i)6-s + (−0.5 − 0.866i)7-s + (0.669 + 0.743i)8-s + (0.961 − 0.275i)9-s + (0.913 − 0.406i)11-s + (0.913 + 0.406i)12-s + (−0.719 − 0.694i)13-s + (−0.241 − 0.970i)14-s + (0.438 + 0.898i)16-s + (0.0348 + 0.999i)17-s + 18-s + (−0.615 − 0.788i)21-s + (0.990 − 0.139i)22-s + ⋯
L(s)  = 1  + (0.961 + 0.275i)2-s + (0.990 − 0.139i)3-s + (0.848 + 0.529i)4-s + (0.990 + 0.139i)6-s + (−0.5 − 0.866i)7-s + (0.669 + 0.743i)8-s + (0.961 − 0.275i)9-s + (0.913 − 0.406i)11-s + (0.913 + 0.406i)12-s + (−0.719 − 0.694i)13-s + (−0.241 − 0.970i)14-s + (0.438 + 0.898i)16-s + (0.0348 + 0.999i)17-s + 18-s + (−0.615 − 0.788i)21-s + (0.990 − 0.139i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0576i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0576i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $0.998 + 0.0576i$
Analytic conductor: \(2.20589\)
Root analytic conductor: \(2.20589\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (446, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 475,\ (0:\ ),\ 0.998 + 0.0576i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.367907878 + 0.09718019789i\)
\(L(\frac12)\) \(\approx\) \(3.367907878 + 0.09718019789i\)
\(L(1)\) \(\approx\) \(2.423982070 + 0.1236179477i\)
\(L(1)\) \(\approx\) \(2.423982070 + 0.1236179477i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
19 \( 1 \)
good2 \( 1 + (0.961 + 0.275i)T \)
3 \( 1 + (0.990 - 0.139i)T \)
7 \( 1 + (-0.5 - 0.866i)T \)
11 \( 1 + (0.913 - 0.406i)T \)
13 \( 1 + (-0.719 - 0.694i)T \)
17 \( 1 + (0.0348 + 0.999i)T \)
23 \( 1 + (-0.997 + 0.0697i)T \)
29 \( 1 + (0.0348 - 0.999i)T \)
31 \( 1 + (-0.978 + 0.207i)T \)
37 \( 1 + (-0.809 + 0.587i)T \)
41 \( 1 + (0.438 + 0.898i)T \)
43 \( 1 + (0.766 + 0.642i)T \)
47 \( 1 + (0.0348 - 0.999i)T \)
53 \( 1 + (0.848 + 0.529i)T \)
59 \( 1 + (0.559 - 0.829i)T \)
61 \( 1 + (-0.997 + 0.0697i)T \)
67 \( 1 + (-0.615 + 0.788i)T \)
71 \( 1 + (-0.374 + 0.927i)T \)
73 \( 1 + (-0.719 + 0.694i)T \)
79 \( 1 + (0.990 - 0.139i)T \)
83 \( 1 + (-0.978 + 0.207i)T \)
89 \( 1 + (0.438 - 0.898i)T \)
97 \( 1 + (-0.615 - 0.788i)T \)
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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

−24.072790768017155160653570555319, −22.56813888188848796756936701263, −22.162557501041487601880119980623, −21.33266044600726083632688194464, −20.467024725549314518736724849663, −19.65899769897401996476241960717, −19.14228455710943111759609106748, −18.11000384245336084120078798744, −16.46194028430365155921139623647, −15.86798596967674993642697420560, −14.86530881900598179961201068671, −14.3321833545809605569110432489, −13.55041649502436602273816222469, −12.32595406568481984816101954551, −12.0629424966966436753193447359, −10.64086470145844778608638420694, −9.4839664163828505296362569742, −9.0768137344225304680762542372, −7.45107742161681590155716081404, −6.76118891765643591065910654603, −5.5212339156288306461150601355, −4.43394338303250573806082845471, −3.562057463010790398162709642086, −2.5170306258488624231223449743, −1.77681216165633993428956645270, 1.50477190303074012426238605876, 2.76725176783653293937763245474, 3.7299077290201284913895256069, 4.28686513003252828572843727054, 5.83114400494392154013405186962, 6.77097013528425622030547084618, 7.60966341565346328497411056127, 8.42683725809734121984028259599, 9.74657805982270889787755448502, 10.59970476439943450557553051965, 11.91859784322842887678771674828, 12.80483397698279214428580069937, 13.48575995472114246912154452379, 14.30879019289030139803659086958, 14.90742927671938863629407898930, 15.87680645762233929094289789314, 16.775045955611755651720794007817, 17.60694958336447261148910348045, 19.16678642423014145261233126613, 19.83744202641803317125046026080, 20.29219455710117840274712068688, 21.42580201728133360579735970463, 22.09507863411967904053578647028, 22.99974974447234591900760558972, 23.97949333901658133757340537330

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