Properties

Label 2-3800-5.4-c1-0-64
Degree $2$
Conductor $3800$
Sign $-0.447 + 0.894i$
Analytic cond. $30.3431$
Root an. cond. $5.50846$
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  − 0.656i·3-s − 0.656i·7-s + 2.56·9-s − 0.343·11-s − 1.91i·13-s − 4.48i·17-s + 19-s − 0.431·21-s − 3.56i·23-s − 3.65i·27-s − 7.99·29-s + 5.73·31-s + 0.225i·33-s + 4.16i·37-s − 1.25·39-s + ⋯
L(s)  = 1  − 0.379i·3-s − 0.248i·7-s + 0.856·9-s − 0.103·11-s − 0.530i·13-s − 1.08i·17-s + 0.229·19-s − 0.0940·21-s − 0.744i·23-s − 0.703i·27-s − 1.48·29-s + 1.03·31-s + 0.0392i·33-s + 0.685i·37-s − 0.201·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3800\)    =    \(2^{3} \cdot 5^{2} \cdot 19\)
Sign: $-0.447 + 0.894i$
Analytic conductor: \(30.3431\)
Root analytic conductor: \(5.50846\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3800} (3649, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3800,\ (\ :1/2),\ -0.447 + 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.587021148\)
\(L(\frac12)\) \(\approx\) \(1.587021148\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
19 \( 1 - T \)
good3 \( 1 + 0.656iT - 3T^{2} \)
7 \( 1 + 0.656iT - 7T^{2} \)
11 \( 1 + 0.343T + 11T^{2} \)
13 \( 1 + 1.91iT - 13T^{2} \)
17 \( 1 + 4.48iT - 17T^{2} \)
23 \( 1 + 3.56iT - 23T^{2} \)
29 \( 1 + 7.99T + 29T^{2} \)
31 \( 1 - 5.73T + 31T^{2} \)
37 \( 1 - 4.16iT - 37T^{2} \)
41 \( 1 + 9.08T + 41T^{2} \)
43 \( 1 - 3.51iT - 43T^{2} \)
47 \( 1 + 3.40iT - 47T^{2} \)
53 \( 1 + 1.68iT - 53T^{2} \)
59 \( 1 - 7.82T + 59T^{2} \)
61 \( 1 + 12.4T + 61T^{2} \)
67 \( 1 + 9.48iT - 67T^{2} \)
71 \( 1 + 9.96T + 71T^{2} \)
73 \( 1 - 7.53iT - 73T^{2} \)
79 \( 1 - 14.9T + 79T^{2} \)
83 \( 1 + 10.9iT - 83T^{2} \)
89 \( 1 - 4.19T + 89T^{2} \)
97 \( 1 + 1.73iT - 97T^{2} \)
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

−8.112964885168464244104191690574, −7.47634757340038166537035919684, −6.90396263227852816107272314130, −6.15930253904647669945888421527, −5.16466178950363103819083318308, −4.53496222725466273279326686745, −3.55297171408828834017597842906, −2.63909753318428756730844710575, −1.57440518974004092979092703762, −0.47451669956612729669134553388, 1.34930806915059113116715711415, 2.21941951284794359741144319074, 3.52026849561204917171305787522, 4.05181969328075046486303025632, 4.95004410902713635860150955305, 5.71464057947804517782289088387, 6.52144299507814607279830446923, 7.32592192071415454664632087420, 7.935185370146756678814597324560, 8.922112117753794363410118227625

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