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Properties

Label 2-475-95.18-c1-0-6
Degree $2$
Conductor $475$
Sign $-0.994 + 0.101i$
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  + (1.61 + 1.61i)2-s + (−1.11 + 1.11i)3-s + 3.21i·4-s − 3.59·6-s + (−1.21 + 1.21i)7-s + (−1.96 + 1.96i)8-s + 0.525i·9-s − 3.52·11-s + (−3.57 − 3.57i)12-s + (1.11 − 1.11i)13-s − 3.92·14-s + 0.0967·16-s + (−3.90 + 3.90i)17-s + (−0.848 + 0.848i)18-s + (3.92 + 1.90i)19-s + ⋯
L(s)  = 1  + (1.14 + 1.14i)2-s + (−0.642 + 0.642i)3-s + 1.60i·4-s − 1.46·6-s + (−0.458 + 0.458i)7-s + (−0.693 + 0.693i)8-s + 0.175i·9-s − 1.06·11-s + (−1.03 − 1.03i)12-s + (0.308 − 0.308i)13-s − 1.04·14-s + 0.0241·16-s + (−0.946 + 0.946i)17-s + (−0.199 + 0.199i)18-s + (0.899 + 0.436i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.994 + 0.101i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.994 + 0.101i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $-0.994 + 0.101i$
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: \((2,\ 475,\ (\ :1/2),\ -0.994 + 0.101i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0874345 - 1.71749i\)
\(L(\frac12)\) \(\approx\) \(0.0874345 - 1.71749i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.92 - 1.90i)T \)
good2 \( 1 + (-1.61 - 1.61i)T + 2iT^{2} \)
3 \( 1 + (1.11 - 1.11i)T - 3iT^{2} \)
7 \( 1 + (1.21 - 1.21i)T - 7iT^{2} \)
11 \( 1 + 3.52T + 11T^{2} \)
13 \( 1 + (-1.11 + 1.11i)T - 13iT^{2} \)
17 \( 1 + (3.90 - 3.90i)T - 17iT^{2} \)
23 \( 1 + (1.21 + 1.21i)T + 23iT^{2} \)
29 \( 1 - 8.68T + 29T^{2} \)
31 \( 1 + 5.14iT - 31T^{2} \)
37 \( 1 + (-2.11 - 2.11i)T + 37iT^{2} \)
41 \( 1 - 11.3iT - 41T^{2} \)
43 \( 1 + (-1.96 - 1.96i)T + 43iT^{2} \)
47 \( 1 + (-5.83 + 5.83i)T - 47iT^{2} \)
53 \( 1 + (-0.107 + 0.107i)T - 53iT^{2} \)
59 \( 1 + 11.3T + 59T^{2} \)
61 \( 1 - 9.13T + 61T^{2} \)
67 \( 1 + (-6.56 - 6.56i)T + 67iT^{2} \)
71 \( 1 + 2.70iT - 71T^{2} \)
73 \( 1 + (4.65 + 4.65i)T + 73iT^{2} \)
79 \( 1 + 3.54T + 79T^{2} \)
83 \( 1 + (-3.06 - 3.06i)T + 83iT^{2} \)
89 \( 1 - 6.24T + 89T^{2} \)
97 \( 1 + (1.42 + 1.42i)T + 97iT^{2} \)
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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

−11.61142518076408545719412349526, −10.58427064361068588749035960230, −9.864389335239218490339466680909, −8.381718338094354049830273842806, −7.69993334184249362231428132627, −6.37213289124562590000869758079, −5.83374918212083595400894770732, −4.96290238427645392339538707177, −4.17710868881550003278389300833, −2.83258955298845706409733335554, 0.796660393342935336524064328510, 2.41538800589178504183820695628, 3.48622837549928553892867199681, 4.70351734566263871679354725086, 5.55574951200508801559967340431, 6.61075970218993429824364363916, 7.46685813818085308235583423323, 9.025905498503783400244913338714, 10.13349016266980527854276132638, 10.88163173267432529773942852398

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