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Properties

Label 2-475-95.37-c1-0-21
Degree $2$
Conductor $475$
Sign $-0.525 + 0.850i$
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  + 2i·4-s + (−3.67 − 3.67i)7-s − 3i·9-s − 4.35·11-s − 4·16-s + (1.32 + 1.32i)17-s − 4.35i·19-s + (2.35 − 2.35i)23-s + (7.35 − 7.35i)28-s + 6·36-s + (−6.03 + 6.03i)43-s − 8.71i·44-s + (−8.67 − 8.67i)47-s + 20.0i·49-s − 4.35·61-s + ⋯
L(s)  = 1  + i·4-s + (−1.39 − 1.39i)7-s − i·9-s − 1.31·11-s − 16-s + (0.320 + 0.320i)17-s − 0.999i·19-s + (0.491 − 0.491i)23-s + (1.39 − 1.39i)28-s + 36-s + (−0.920 + 0.920i)43-s − 1.31i·44-s + (−1.26 − 1.26i)47-s + 2.86i·49-s − 0.558·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.525 + 0.850i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.525 + 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $-0.525 + 0.850i$
Analytic conductor: \(3.79289\)
Root analytic conductor: \(1.94753\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (132, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 475,\ (\ :1/2),\ -0.525 + 0.850i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.260533 - 0.467294i\)
\(L(\frac12)\) \(\approx\) \(0.260533 - 0.467294i\)
\(L(\frac{3}{2})\) 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 \( 1 + 4.35iT \)
good2 \( 1 - 2iT^{2} \)
3 \( 1 + 3iT^{2} \)
7 \( 1 + (3.67 + 3.67i)T + 7iT^{2} \)
11 \( 1 + 4.35T + 11T^{2} \)
13 \( 1 + 13iT^{2} \)
17 \( 1 + (-1.32 - 1.32i)T + 17iT^{2} \)
23 \( 1 + (-2.35 + 2.35i)T - 23iT^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 - 37iT^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 + (6.03 - 6.03i)T - 43iT^{2} \)
47 \( 1 + (8.67 + 8.67i)T + 47iT^{2} \)
53 \( 1 + 53iT^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 + 4.35T + 61T^{2} \)
67 \( 1 - 67iT^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 + (1.03 - 1.03i)T - 73iT^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + (-12.3 + 12.3i)T - 83iT^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 - 97iT^{2} \)
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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

−10.60086460262943919192949000977, −9.865790766024906865822008375870, −8.941193147482263649774053666819, −7.84509330931984383084080124554, −7.01987432224481562987139020813, −6.37954543982250408774241704807, −4.70778975654487276188941393264, −3.56657949847332998137500340845, −2.94521458717871129049045391629, −0.30374544067046084425580504143, 2.09818360877445881677812392802, 3.10852318202880776455356011934, 5.05351519780631410895771827121, 5.58508075548018664640411316183, 6.44941810587166817618596429198, 7.70708095785568333377653997360, 8.764032885773144063008072124167, 9.731342893085473341629824212403, 10.22344943577430790637512610936, 11.16617735249840585536003821803

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