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Properties

Label 2-475-95.18-c1-0-13
Degree $2$
Conductor $475$
Sign $-0.339 + 0.940i$
Analytic cond. $3.79289$
Root an. cond. $1.94753$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.10 − 1.10i)2-s + (−1.29 + 1.29i)3-s + 0.460i·4-s + 2.87·6-s + (1.53 − 1.53i)7-s + (−1.70 + 1.70i)8-s − 0.369i·9-s − 2.63·11-s + (−0.598 − 0.598i)12-s + (1.29 − 1.29i)13-s − 3.41·14-s + 4.70·16-s + (0.709 − 0.709i)17-s + (−0.409 + 0.409i)18-s + (3.41 − 2.70i)19-s + ⋯
L(s)  = 1  + (−0.784 − 0.784i)2-s + (−0.749 + 0.749i)3-s + 0.230i·4-s + 1.17·6-s + (0.581 − 0.581i)7-s + (−0.603 + 0.603i)8-s − 0.123i·9-s − 0.793·11-s + (−0.172 − 0.172i)12-s + (0.359 − 0.359i)13-s − 0.912·14-s + 1.17·16-s + (0.172 − 0.172i)17-s + (−0.0965 + 0.0965i)18-s + (0.783 − 0.621i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.339 + 0.940i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.339 + 0.940i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $-0.339 + 0.940i$
Analytic conductor: \(3.79289\)
Root analytic conductor: \(1.94753\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (18, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 475,\ (\ :1/2),\ -0.339 + 0.940i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.325944 - 0.464229i\)
\(L(\frac12)\) \(\approx\) \(0.325944 - 0.464229i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
19 \( 1 + (-3.41 + 2.70i)T \)
good2 \( 1 + (1.10 + 1.10i)T + 2iT^{2} \)
3 \( 1 + (1.29 - 1.29i)T - 3iT^{2} \)
7 \( 1 + (-1.53 + 1.53i)T - 7iT^{2} \)
11 \( 1 + 2.63T + 11T^{2} \)
13 \( 1 + (-1.29 + 1.29i)T - 13iT^{2} \)
17 \( 1 + (-0.709 + 0.709i)T - 17iT^{2} \)
23 \( 1 + (-1.53 - 1.53i)T + 23iT^{2} \)
29 \( 1 + 1.84T + 29T^{2} \)
31 \( 1 + 10.8iT - 31T^{2} \)
37 \( 1 + (3.51 + 3.51i)T + 37iT^{2} \)
41 \( 1 + 5.83iT - 41T^{2} \)
43 \( 1 + (8.21 + 8.21i)T + 43iT^{2} \)
47 \( 1 + (-6.80 + 6.80i)T - 47iT^{2} \)
53 \( 1 + (-6.11 + 6.11i)T - 53iT^{2} \)
59 \( 1 - 5.83T + 59T^{2} \)
61 \( 1 + 10.2T + 61T^{2} \)
67 \( 1 + (-1.67 - 1.67i)T + 67iT^{2} \)
71 \( 1 - 3.99iT - 71T^{2} \)
73 \( 1 + (-7.38 - 7.38i)T + 73iT^{2} \)
79 \( 1 - 12.6T + 79T^{2} \)
83 \( 1 + (2.51 + 2.51i)T + 83iT^{2} \)
89 \( 1 + 16.6T + 89T^{2} \)
97 \( 1 + (-9.14 - 9.14i)T + 97iT^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.65677170858773408638333909223, −10.14618141540594095849499542797, −9.288059383291770290654961231612, −8.218863371725385824106534434283, −7.30706398840152818613063617251, −5.62418441003877487770996856262, −5.15969130942205933717101282262, −3.78229744774062157215098114197, −2.24567092219684246859713393160, −0.54264711455239919316528614581, 1.33982368712425331839247688337, 3.23188006333097075240491166034, 5.05614795374867816997166177874, 5.97716109904758782266594149130, 6.77189055019869436988510570533, 7.64152466003471993551792597443, 8.380026785271997512933639544016, 9.221538255554421420950231582254, 10.31236907636063033303769108515, 11.37153374876844443795129824134

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