Properties

Label 2-475-95.18-c1-0-26
Degree $2$
Conductor $475$
Sign $-0.826 + 0.563i$
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  + (−0.813 − 0.813i)2-s + (2.01 − 2.01i)3-s − 0.675i·4-s − 3.28·6-s + (2.67 − 2.67i)7-s + (−2.17 + 2.17i)8-s − 5.15i·9-s + 2.15·11-s + (−1.36 − 1.36i)12-s + (−2.01 + 2.01i)13-s − 4.35·14-s + 2.19·16-s + (−1.80 + 1.80i)17-s + (−4.19 + 4.19i)18-s + (4.35 − 0.193i)19-s + ⋯
L(s)  = 1  + (−0.575 − 0.575i)2-s + (1.16 − 1.16i)3-s − 0.337i·4-s − 1.34·6-s + (1.01 − 1.01i)7-s + (−0.769 + 0.769i)8-s − 1.71i·9-s + 0.650·11-s + (−0.393 − 0.393i)12-s + (−0.560 + 0.560i)13-s − 1.16·14-s + 0.548·16-s + (−0.438 + 0.438i)17-s + (−0.989 + 0.989i)18-s + (0.999 − 0.0444i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.826 + 0.563i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.826 + 0.563i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $-0.826 + 0.563i$
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: \((2,\ 475,\ (\ :1/2),\ -0.826 + 0.563i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.481189 - 1.56085i\)
\(L(\frac12)\) \(\approx\) \(0.481189 - 1.56085i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-4.35 + 0.193i)T \)
good2 \( 1 + (0.813 + 0.813i)T + 2iT^{2} \)
3 \( 1 + (-2.01 + 2.01i)T - 3iT^{2} \)
7 \( 1 + (-2.67 + 2.67i)T - 7iT^{2} \)
11 \( 1 - 2.15T + 11T^{2} \)
13 \( 1 + (2.01 - 2.01i)T - 13iT^{2} \)
17 \( 1 + (1.80 - 1.80i)T - 17iT^{2} \)
23 \( 1 + (-2.67 - 2.67i)T + 23iT^{2} \)
29 \( 1 + 7.29T + 29T^{2} \)
31 \( 1 - 2.09iT - 31T^{2} \)
37 \( 1 + (-0.391 - 0.391i)T + 37iT^{2} \)
41 \( 1 - 3.51iT - 41T^{2} \)
43 \( 1 + (-5.24 - 5.24i)T + 43iT^{2} \)
47 \( 1 + (1.63 - 1.63i)T - 47iT^{2} \)
53 \( 1 + (4.43 - 4.43i)T - 53iT^{2} \)
59 \( 1 + 3.51T + 59T^{2} \)
61 \( 1 + 4.93T + 61T^{2} \)
67 \( 1 + (7.68 + 7.68i)T + 67iT^{2} \)
71 \( 1 + 10.8iT - 71T^{2} \)
73 \( 1 + (9.73 + 9.73i)T + 73iT^{2} \)
79 \( 1 - 5.19T + 79T^{2} \)
83 \( 1 + (-8.44 - 8.44i)T + 83iT^{2} \)
89 \( 1 - 5.60T + 89T^{2} \)
97 \( 1 + (-5.59 - 5.59i)T + 97iT^{2} \)
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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

−10.72557965238317958254209478882, −9.422103457152305801930562463699, −9.048028120037029062052143175303, −7.84909100617430221580179380836, −7.39730066055765411756777799854, −6.27663108769124582801573011898, −4.71199655715028510400329280736, −3.26652989929715810538784665368, −1.88420899210643139109019360402, −1.22664173459353405325987202266, 2.40828679500271495065038941645, 3.44051367281496132874004090384, 4.58088755344374605998687000416, 5.62567215439025953977828293204, 7.25233650992303655073187587791, 7.977974338496672757190988157560, 8.906682618928844393097274426903, 9.122958555510987797688112286982, 10.05538654438885118860143138402, 11.27723745713784457912881964656

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