Properties

Label 2-475-95.94-c2-0-34
Degree $2$
Conductor $475$
Sign $-0.0838 + 0.996i$
Analytic cond. $12.9428$
Root an. cond. $3.59761$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2.34·2-s − 5.73·3-s + 1.51·4-s − 13.4·6-s + 9.56i·7-s − 5.83·8-s + 23.9·9-s + 4.15·11-s − 8.68·12-s − 4.55·13-s + 22.4i·14-s − 19.7·16-s − 27.9i·17-s + 56.1·18-s + (−7.04 − 17.6i)19-s + ⋯
L(s)  = 1  + 1.17·2-s − 1.91·3-s + 0.378·4-s − 2.24·6-s + 1.36i·7-s − 0.729·8-s + 2.65·9-s + 0.377·11-s − 0.723·12-s − 0.350·13-s + 1.60i·14-s − 1.23·16-s − 1.64i·17-s + 3.11·18-s + (−0.370 − 0.928i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0838 + 0.996i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0838 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $-0.0838 + 0.996i$
Analytic conductor: \(12.9428\)
Root analytic conductor: \(3.59761\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (474, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 475,\ (\ :1),\ -0.0838 + 0.996i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.7699449931\)
\(L(\frac12)\) \(\approx\) \(0.7699449931\)
\(L(2)\) 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 \( 1 + (7.04 + 17.6i)T \)
good2 \( 1 - 2.34T + 4T^{2} \)
3 \( 1 + 5.73T + 9T^{2} \)
7 \( 1 - 9.56iT - 49T^{2} \)
11 \( 1 - 4.15T + 121T^{2} \)
13 \( 1 + 4.55T + 169T^{2} \)
17 \( 1 + 27.9iT - 289T^{2} \)
23 \( 1 - 13.9iT - 529T^{2} \)
29 \( 1 + 14.6iT - 841T^{2} \)
31 \( 1 + 28.9iT - 961T^{2} \)
37 \( 1 - 30.8T + 1.36e3T^{2} \)
41 \( 1 + 44.2iT - 1.68e3T^{2} \)
43 \( 1 + 69.2iT - 1.84e3T^{2} \)
47 \( 1 - 30.1iT - 2.20e3T^{2} \)
53 \( 1 + 54.3T + 2.80e3T^{2} \)
59 \( 1 - 31.3iT - 3.48e3T^{2} \)
61 \( 1 - 99.9T + 3.72e3T^{2} \)
67 \( 1 + 3.05T + 4.48e3T^{2} \)
71 \( 1 - 23.0iT - 5.04e3T^{2} \)
73 \( 1 + 21.1iT - 5.32e3T^{2} \)
79 \( 1 - 77.7iT - 6.24e3T^{2} \)
83 \( 1 + 93.1iT - 6.88e3T^{2} \)
89 \( 1 + 126. iT - 7.92e3T^{2} \)
97 \( 1 - 77.2T + 9.40e3T^{2} \)
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

−11.17496216125404633533607783188, −9.747070779883888504260658624568, −9.071716602254787131192046571817, −7.25120538837762670570562942843, −6.37442486960088555776130936442, −5.55898295999934421666583516674, −5.09763079744168741637301305512, −4.17336452513676768194999205233, −2.47406701086633123939018632866, −0.30219248620013287892202599633, 1.26469490719596338768522665166, 3.80064002533575368437758934808, 4.38441146407525277714004382232, 5.25429485362102600878589507959, 6.35016199860357657714387445818, 6.63512567469698832857336636442, 7.989380921841623714679515753335, 9.742746333649602529393067127697, 10.52918031988334388308718367332, 11.12647787281824765358511662659

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