↑

Properties

Label 2-1900-95.68-c0-0-0
Degree $2$
Conductor $1900$
Sign $0.960 + 0.277i$
Analytic cond. $0.948223$
Root an. cond. $0.973767$
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  + (1.36 − 0.366i)3-s + (0.866 − 0.5i)9-s + 11-s + (−0.366 + 1.36i)13-s + (−0.366 − 1.36i)17-s + (0.866 − 0.5i)19-s + (−0.866 + 0.5i)29-s − 31-s + (1.36 − 0.366i)33-s + 2i·39-s + (1.36 + 0.366i)47-s + i·49-s + (−1 − 1.73i)51-s + (0.366 − 1.36i)53-s + (0.999 − i)57-s + ⋯
L(s)  = 1  + (1.36 − 0.366i)3-s + (0.866 − 0.5i)9-s + 11-s + (−0.366 + 1.36i)13-s + (−0.366 − 1.36i)17-s + (0.866 − 0.5i)19-s + (−0.866 + 0.5i)29-s − 31-s + (1.36 − 0.366i)33-s + 2i·39-s + (1.36 + 0.366i)47-s + i·49-s + (−1 − 1.73i)51-s + (0.366 − 1.36i)53-s + (0.999 − i)57-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.960 + 0.277i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.960 + 0.277i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1900\)    =    \(2^{2} \cdot 5^{2} \cdot 19\)
Sign: $0.960 + 0.277i$
Analytic conductor: \(0.948223\)
Root analytic conductor: \(0.973767\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1900} (1493, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1900,\ (\ :0),\ 0.960 + 0.277i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.841652886\)
\(L(\frac12)\) \(\approx\) \(1.841652886\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
19 \( 1 + (-0.866 + 0.5i)T \)
good3 \( 1 + (-1.36 + 0.366i)T + (0.866 - 0.5i)T^{2} \)
7 \( 1 - iT^{2} \)
11 \( 1 - T + T^{2} \)
13 \( 1 + (0.366 - 1.36i)T + (-0.866 - 0.5i)T^{2} \)
17 \( 1 + (0.366 + 1.36i)T + (-0.866 + 0.5i)T^{2} \)
23 \( 1 + (-0.866 - 0.5i)T^{2} \)
29 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
31 \( 1 + T + T^{2} \)
37 \( 1 - iT^{2} \)
41 \( 1 + (-0.5 - 0.866i)T^{2} \)
43 \( 1 + (0.866 - 0.5i)T^{2} \)
47 \( 1 + (-1.36 - 0.366i)T + (0.866 + 0.5i)T^{2} \)
53 \( 1 + (-0.366 + 1.36i)T + (-0.866 - 0.5i)T^{2} \)
59 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (1.36 + 0.366i)T + (0.866 + 0.5i)T^{2} \)
71 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
73 \( 1 + (0.866 - 0.5i)T^{2} \)
79 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
83 \( 1 + iT^{2} \)
89 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
97 \( 1 + (-0.866 + 0.5i)T^{2} \)
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

−9.186000993643986013578594842835, −8.919121741817829198595992592193, −7.65421144829346351958153048841, −7.22408574963719423001656699894, −6.52546200645533327677722168792, −5.21908947805727914539195924232, −4.23941717552388011520151278704, −3.38574206943600046849459141704, −2.45231505569630955189153346376, −1.53806903695440170300265687993, 1.59155617050334480192927833145, 2.70918433544033429040583093569, 3.62652131941826562328968734346, 4.11426053011071020663282230444, 5.44891611792543243254567145108, 6.20571139395083512838391271242, 7.54287177936144695084707676346, 7.79664069975286063727612546088, 8.964648367930517526376842851956, 9.062530216394892608241402247292

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