↑

Properties

Label 2-3800-760.539-c0-0-5
Degree $2$
Conductor $3800$
Sign $-0.152 + 0.988i$
Analytic cond. $1.89644$
Root an. cond. $1.37711$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.866 − 0.5i)2-s + (0.866 − 0.5i)3-s + (0.499 − 0.866i)4-s + (0.499 − 0.866i)6-s − 0.999i·8-s − 11-s − 0.999i·12-s + (−0.5 − 0.866i)16-s + (1.73 − i)17-s + (0.5 − 0.866i)19-s + (−0.866 + 0.5i)22-s + (−0.5 − 0.866i)24-s + i·27-s + (−0.866 − 0.499i)32-s + (−0.866 + 0.5i)33-s + (0.999 − 1.73i)34-s + ⋯
L(s)  = 1  + (0.866 − 0.5i)2-s + (0.866 − 0.5i)3-s + (0.499 − 0.866i)4-s + (0.499 − 0.866i)6-s − 0.999i·8-s − 11-s − 0.999i·12-s + (−0.5 − 0.866i)16-s + (1.73 − i)17-s + (0.5 − 0.866i)19-s + (−0.866 + 0.5i)22-s + (−0.5 − 0.866i)24-s + i·27-s + (−0.866 − 0.499i)32-s + (−0.866 + 0.5i)33-s + (0.999 − 1.73i)34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.152 + 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.152 + 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3800\)    =    \(2^{3} \cdot 5^{2} \cdot 19\)
Sign: $-0.152 + 0.988i$
Analytic conductor: \(1.89644\)
Root analytic conductor: \(1.37711\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3800} (1299, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3800,\ (\ :0),\ -0.152 + 0.988i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.778594357\)
\(L(\frac12)\) \(\approx\) \(2.778594357\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 + 0.5i)T \)
5 \( 1 \)
19 \( 1 + (-0.5 + 0.866i)T \)
good3 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
7 \( 1 + T^{2} \)
11 \( 1 + T + T^{2} \)
13 \( 1 + (-0.5 - 0.866i)T^{2} \)
17 \( 1 + (-1.73 + i)T + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.5 - 0.866i)T^{2} \)
29 \( 1 + (0.5 + 0.866i)T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
43 \( 1 + (1.73 - i)T + (0.5 - 0.866i)T^{2} \)
47 \( 1 + (-0.5 - 0.866i)T^{2} \)
53 \( 1 + (-0.5 - 0.866i)T^{2} \)
59 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
71 \( 1 + (0.5 - 0.866i)T^{2} \)
73 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 - iT - T^{2} \)
89 \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \)
97 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
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

−8.262057438386319443406924296115, −7.71270432103438642481715771105, −7.09569751583113792543885483726, −6.17239023837203753593964451194, −5.08583679416755380118742700208, −4.96694254318839599471162262452, −3.42188949055513843544158764582, −3.01080050816064212321032002398, −2.26224199376516917850708167908, −1.13594859349304017073477690312, 1.83794377455359326389843473585, 2.97274488362386089879285307783, 3.46929155713256913925837996333, 4.14042401226337320820489472849, 5.26571185475875722080018992640, 5.67209296982242134807563279161, 6.58360832723917082814862396437, 7.60193518258426717272707781297, 8.070434135336012262090176231463, 8.543548004661887693051885061789

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