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Properties

Label 2-475-19.4-c1-0-20
Degree $2$
Conductor $475$
Sign $0.823 + 0.566i$
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  + (0.984 + 0.358i)2-s + (0.0922 + 0.523i)3-s + (−0.691 − 0.580i)4-s + (−0.0966 + 0.548i)6-s + (1.37 − 2.37i)7-s + (−1.52 − 2.63i)8-s + (2.55 − 0.929i)9-s + (−0.416 − 0.721i)11-s + (0.239 − 0.415i)12-s + (0.106 − 0.601i)13-s + (2.19 − 1.84i)14-s + (−0.239 − 1.35i)16-s + (−4.54 − 1.65i)17-s + 2.84·18-s + (4.35 + 0.175i)19-s + ⋯
L(s)  = 1  + (0.695 + 0.253i)2-s + (0.0532 + 0.302i)3-s + (−0.345 − 0.290i)4-s + (−0.0394 + 0.223i)6-s + (0.517 − 0.896i)7-s + (−0.537 − 0.930i)8-s + (0.851 − 0.309i)9-s + (−0.125 − 0.217i)11-s + (0.0692 − 0.119i)12-s + (0.0294 − 0.166i)13-s + (0.587 − 0.493i)14-s + (−0.0598 − 0.339i)16-s + (−1.10 − 0.401i)17-s + 0.670·18-s + (0.999 + 0.0402i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.823 + 0.566i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.823 + 0.566i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $0.823 + 0.566i$
Analytic conductor: \(3.79289\)
Root analytic conductor: \(1.94753\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 475,\ (\ :1/2),\ 0.823 + 0.566i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.81700 - 0.564855i\)
\(L(\frac12)\) \(\approx\) \(1.81700 - 0.564855i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-4.35 - 0.175i)T \)
good2 \( 1 + (-0.984 - 0.358i)T + (1.53 + 1.28i)T^{2} \)
3 \( 1 + (-0.0922 - 0.523i)T + (-2.81 + 1.02i)T^{2} \)
7 \( 1 + (-1.37 + 2.37i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (0.416 + 0.721i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.106 + 0.601i)T + (-12.2 - 4.44i)T^{2} \)
17 \( 1 + (4.54 + 1.65i)T + (13.0 + 10.9i)T^{2} \)
23 \( 1 + (-2.87 - 2.41i)T + (3.99 + 22.6i)T^{2} \)
29 \( 1 + (-3.73 + 1.35i)T + (22.2 - 18.6i)T^{2} \)
31 \( 1 + (-3.46 + 5.99i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 4.33T + 37T^{2} \)
41 \( 1 + (-0.923 - 5.23i)T + (-38.5 + 14.0i)T^{2} \)
43 \( 1 + (8.01 - 6.72i)T + (7.46 - 42.3i)T^{2} \)
47 \( 1 + (3.19 - 1.16i)T + (36.0 - 30.2i)T^{2} \)
53 \( 1 + (10.4 + 8.78i)T + (9.20 + 52.1i)T^{2} \)
59 \( 1 + (9.41 + 3.42i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (-6.94 - 5.83i)T + (10.5 + 60.0i)T^{2} \)
67 \( 1 + (10.2 - 3.73i)T + (51.3 - 43.0i)T^{2} \)
71 \( 1 + (0.519 - 0.435i)T + (12.3 - 69.9i)T^{2} \)
73 \( 1 + (-1.21 - 6.90i)T + (-68.5 + 24.9i)T^{2} \)
79 \( 1 + (-0.604 - 3.42i)T + (-74.2 + 27.0i)T^{2} \)
83 \( 1 + (-2.48 + 4.30i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (1.02 - 5.79i)T + (-83.6 - 30.4i)T^{2} \)
97 \( 1 + (-11.6 - 4.25i)T + (74.3 + 62.3i)T^{2} \)
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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

−10.93658607908343008808271919024, −9.878995556154130634248861970428, −9.433720381833299908060980623113, −8.097021715444501111427423870821, −7.07899759224982467592148946230, −6.20450995816354933056767188151, −4.85895898185066584787256595623, −4.41728303947476092353253092988, −3.26632650190679473516697304540, −1.06643899827237836584536742837, 1.91098695160776325855651381957, 3.08657903618750868653189602953, 4.53155038980056720457084221296, 5.04733540303347666091191810349, 6.34455427655429772483991471875, 7.47709461777338048289571020834, 8.477170570827184482398588972285, 9.097420350194133953000043229523, 10.34902773014809716441171821020, 11.36592161044742051684409558851

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