↑

Properties

Label 2-1200-3.2-c2-0-67
Degree $2$
Conductor $1200$
Sign $-0.860 + 0.509i$
Analytic cond. $32.6976$
Root an. cond. $5.71818$
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  + (2.58 − 1.52i)3-s − 7.48·7-s + (4.32 − 7.89i)9-s + 8.48i·11-s + 10·13-s − 30.3i·17-s − 26.9·19-s + (−19.3 + 11.4i)21-s − 9.17i·23-s + (−0.905 − 26.9i)27-s + 26.8i·29-s − 8·31-s + (12.9 + 21.9i)33-s − 15.9·37-s + (25.8 − 15.2i)39-s + ⋯
L(s)  = 1  + (0.860 − 0.509i)3-s − 1.06·7-s + (0.480 − 0.876i)9-s + 0.771i·11-s + 0.769·13-s − 1.78i·17-s − 1.41·19-s + (−0.920 + 0.545i)21-s − 0.398i·23-s + (−0.0335 − 0.999i)27-s + 0.925i·29-s − 0.258·31-s + (0.393 + 0.663i)33-s − 0.431·37-s + (0.661 − 0.392i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.860 + 0.509i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.860 + 0.509i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1200\)    =    \(2^{4} \cdot 3 \cdot 5^{2}\)
Sign: $-0.860 + 0.509i$
Analytic conductor: \(32.6976\)
Root analytic conductor: \(5.71818\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1200} (401, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1200,\ (\ :1),\ -0.860 + 0.509i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.245025118\)
\(L(\frac12)\) \(\approx\) \(1.245025118\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-2.58 + 1.52i)T \)
5 \( 1 \)
good7 \( 1 + 7.48T + 49T^{2} \)
11 \( 1 - 8.48iT - 121T^{2} \)
13 \( 1 - 10T + 169T^{2} \)
17 \( 1 + 30.3iT - 289T^{2} \)
19 \( 1 + 26.9T + 361T^{2} \)
23 \( 1 + 9.17iT - 529T^{2} \)
29 \( 1 - 26.8iT - 841T^{2} \)
31 \( 1 + 8T + 961T^{2} \)
37 \( 1 + 15.9T + 1.36e3T^{2} \)
41 \( 1 + 47.3iT - 1.68e3T^{2} \)
43 \( 1 + 14.4T + 1.84e3T^{2} \)
47 \( 1 + 45.8iT - 2.20e3T^{2} \)
53 \( 1 + 30.3iT - 2.80e3T^{2} \)
59 \( 1 + 24.0iT - 3.48e3T^{2} \)
61 \( 1 + 53.9T + 3.72e3T^{2} \)
67 \( 1 + 110.T + 4.48e3T^{2} \)
71 \( 1 - 15.5iT - 5.04e3T^{2} \)
73 \( 1 + 87.9T + 5.32e3T^{2} \)
79 \( 1 - 46.9T + 6.24e3T^{2} \)
83 \( 1 + 26.1iT - 6.88e3T^{2} \)
89 \( 1 + 60.7iT - 7.92e3T^{2} \)
97 \( 1 + 36.0T + 9.40e3T^{2} \)
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

−9.051952060715138543546289675367, −8.626158664385862651451175603508, −7.39356482995759398248471022634, −6.89451773521335461913529429910, −6.13760809400984208336137914541, −4.79437497648897878521946018352, −3.71196460909557438232381425155, −2.87899392767149723104521384033, −1.85793307687331360402134432633, −0.30629996211970745164458111073, 1.63908055503996504223069920118, 2.93351135935764387820976155290, 3.69922229334494770067822493574, 4.41396993753707649611116773356, 5.97529433200880994999514172575, 6.33145375050248219925412862348, 7.67032635835143581638923757206, 8.471562146548354081734719734182, 8.944226127075785810854398442628, 9.889195216439294993896203629173

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