↑

Properties

Label 2-475-95.54-c1-0-18
Degree $2$
Conductor $475$
Sign $0.723 - 0.690i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.406 − 0.484i)2-s + (0.656 + 1.80i)3-s + (0.277 + 1.57i)4-s + (1.14 + 0.415i)6-s + (3.54 − 2.04i)7-s + (1.97 + 1.13i)8-s + (−0.524 + 0.440i)9-s + (2.17 − 3.76i)11-s + (−2.65 + 1.53i)12-s + (−0.530 + 1.45i)13-s + (0.449 − 2.54i)14-s + (−1.65 + 0.601i)16-s + (−4.08 + 4.87i)17-s + 0.433i·18-s + (−0.708 − 4.30i)19-s + ⋯
L(s)  = 1  + (0.287 − 0.342i)2-s + (0.379 + 1.04i)3-s + (0.138 + 0.787i)4-s + (0.465 + 0.169i)6-s + (1.33 − 0.772i)7-s + (0.697 + 0.402i)8-s + (−0.174 + 0.146i)9-s + (0.655 − 1.13i)11-s + (−0.767 + 0.443i)12-s + (−0.147 + 0.404i)13-s + (0.120 − 0.681i)14-s + (−0.412 + 0.150i)16-s + (−0.991 + 1.18i)17-s + 0.102i·18-s + (−0.162 − 0.986i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.723 - 0.690i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.723 - 0.690i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $0.723 - 0.690i$
Analytic conductor: \(3.79289\)
Root analytic conductor: \(1.94753\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (149, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 475,\ (\ :1/2),\ 0.723 - 0.690i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.04627 + 0.819508i\)
\(L(\frac12)\) \(\approx\) \(2.04627 + 0.819508i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.708 + 4.30i)T \)
good2 \( 1 + (-0.406 + 0.484i)T + (-0.347 - 1.96i)T^{2} \)
3 \( 1 + (-0.656 - 1.80i)T + (-2.29 + 1.92i)T^{2} \)
7 \( 1 + (-3.54 + 2.04i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-2.17 + 3.76i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.530 - 1.45i)T + (-9.95 - 8.35i)T^{2} \)
17 \( 1 + (4.08 - 4.87i)T + (-2.95 - 16.7i)T^{2} \)
23 \( 1 + (3.31 - 0.583i)T + (21.6 - 7.86i)T^{2} \)
29 \( 1 + (3.99 - 3.35i)T + (5.03 - 28.5i)T^{2} \)
31 \( 1 + (3.28 + 5.68i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 0.180iT - 37T^{2} \)
41 \( 1 + (0.0242 - 0.00881i)T + (31.4 - 26.3i)T^{2} \)
43 \( 1 + (4.50 + 0.793i)T + (40.4 + 14.7i)T^{2} \)
47 \( 1 + (0.919 + 1.09i)T + (-8.16 + 46.2i)T^{2} \)
53 \( 1 + (1.57 - 0.278i)T + (49.8 - 18.1i)T^{2} \)
59 \( 1 + (7.31 + 6.13i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (1.05 + 5.99i)T + (-57.3 + 20.8i)T^{2} \)
67 \( 1 + (-6.60 - 7.87i)T + (-11.6 + 65.9i)T^{2} \)
71 \( 1 + (1.88 - 10.7i)T + (-66.7 - 24.2i)T^{2} \)
73 \( 1 + (4.65 + 12.7i)T + (-55.9 + 46.9i)T^{2} \)
79 \( 1 + (-16.2 + 5.90i)T + (60.5 - 50.7i)T^{2} \)
83 \( 1 + (-4.46 + 2.57i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (0.477 + 0.173i)T + (68.1 + 57.2i)T^{2} \)
97 \( 1 + (2.10 - 2.51i)T + (-16.8 - 95.5i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.10832981821070830157015407708, −10.56548713825252316956528343845, −9.181716156150290452165718423050, −8.543111752064009462225864779152, −7.69747517894737277276539510336, −6.54018796516474493023773138443, −4.89695145194417929099377292890, −4.11876033000691716763478546626, −3.54821120540683478079173746262, −1.90095332762640756864766444667, 1.59137648882268296639924085646, 2.20193762678024254498291763606, 4.44262570198593047059620273407, 5.22735895061768627770358316597, 6.35450458216049069877638937365, 7.24287378168616754246024330258, 7.914142882227456390328752038747, 8.982279622781872063516440461500, 9.958776404902577140550515390589, 11.04289117660249335318207321326

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