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Properties

Label 2-475-19.4-c1-0-19
Degree $2$
Conductor $475$
Sign $0.954 - 0.299i$
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  + (2.18 + 0.795i)2-s + (−0.199 − 1.13i)3-s + (2.60 + 2.18i)4-s + (0.463 − 2.62i)6-s + (−0.0716 + 0.124i)7-s + (1.63 + 2.82i)8-s + (1.57 − 0.574i)9-s + (1.40 + 2.43i)11-s + (1.95 − 3.38i)12-s + (−0.296 + 1.68i)13-s + (−0.255 + 0.214i)14-s + (0.135 + 0.766i)16-s + (3.34 + 1.21i)17-s + 3.90·18-s + (2.82 − 3.32i)19-s + ⋯
L(s)  = 1  + (1.54 + 0.562i)2-s + (−0.115 − 0.653i)3-s + (1.30 + 1.09i)4-s + (0.189 − 1.07i)6-s + (−0.0270 + 0.0469i)7-s + (0.576 + 0.999i)8-s + (0.526 − 0.191i)9-s + (0.424 + 0.735i)11-s + (0.564 − 0.977i)12-s + (−0.0822 + 0.466i)13-s + (−0.0682 + 0.0572i)14-s + (0.0338 + 0.191i)16-s + (0.811 + 0.295i)17-s + 0.920·18-s + (0.647 − 0.762i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.954 - 0.299i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.954 - 0.299i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $0.954 - 0.299i$
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: \((2,\ 475,\ (\ :1/2),\ 0.954 - 0.299i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.18431 + 0.488399i\)
\(L(\frac12)\) \(\approx\) \(3.18431 + 0.488399i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.82 + 3.32i)T \)
good2 \( 1 + (-2.18 - 0.795i)T + (1.53 + 1.28i)T^{2} \)
3 \( 1 + (0.199 + 1.13i)T + (-2.81 + 1.02i)T^{2} \)
7 \( 1 + (0.0716 - 0.124i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.40 - 2.43i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.296 - 1.68i)T + (-12.2 - 4.44i)T^{2} \)
17 \( 1 + (-3.34 - 1.21i)T + (13.0 + 10.9i)T^{2} \)
23 \( 1 + (5.51 + 4.62i)T + (3.99 + 22.6i)T^{2} \)
29 \( 1 + (7.11 - 2.58i)T + (22.2 - 18.6i)T^{2} \)
31 \( 1 + (2.42 - 4.19i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 9.49T + 37T^{2} \)
41 \( 1 + (0.0325 + 0.184i)T + (-38.5 + 14.0i)T^{2} \)
43 \( 1 + (8.42 - 7.06i)T + (7.46 - 42.3i)T^{2} \)
47 \( 1 + (3.44 - 1.25i)T + (36.0 - 30.2i)T^{2} \)
53 \( 1 + (1.25 + 1.05i)T + (9.20 + 52.1i)T^{2} \)
59 \( 1 + (-5.04 - 1.83i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (-7.74 - 6.50i)T + (10.5 + 60.0i)T^{2} \)
67 \( 1 + (-5.64 + 2.05i)T + (51.3 - 43.0i)T^{2} \)
71 \( 1 + (-0.414 + 0.347i)T + (12.3 - 69.9i)T^{2} \)
73 \( 1 + (1.81 + 10.3i)T + (-68.5 + 24.9i)T^{2} \)
79 \( 1 + (2.01 + 11.4i)T + (-74.2 + 27.0i)T^{2} \)
83 \( 1 + (-3.79 + 6.57i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-0.251 + 1.42i)T + (-83.6 - 30.4i)T^{2} \)
97 \( 1 + (-8.27 - 3.01i)T + (74.3 + 62.3i)T^{2} \)
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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

−11.68405573381964794445010643330, −10.26599139732482116515009201978, −9.233020906044950107548540018088, −7.78526309401430126733699689097, −7.01462335779717868073078349609, −6.44719681473159350122936173396, −5.35642745886081386735352298524, −4.39746258441566629843875478180, −3.41600056230304557408212449860, −1.83772118509128862436433890745, 1.83451362306434381417639504589, 3.59433103548878093777117661187, 3.78536569942534180595940269621, 5.34759337943123508821979237225, 5.58625837901607983836822124955, 7.01122215416724557321778920198, 8.182362991562118362202391543321, 9.677418014292928807078234770024, 10.22946074606521410127809057073, 11.28084221947792430044868918188

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