↑

Properties

Label 2-1200-5.3-c2-0-32
Degree $2$
Conductor $1200$
Sign $-0.229 + 0.973i$
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  + (1.22 + 1.22i)3-s + (−0.550 + 0.550i)7-s + 2.99i·9-s + 1.55·11-s + (−9.55 − 9.55i)13-s + (−11.1 + 11.1i)17-s − 12.6i·19-s − 1.34·21-s + (−21.3 − 21.3i)23-s + (−3.67 + 3.67i)27-s − 44.0i·29-s + 44.4·31-s + (1.89 + 1.89i)33-s + (−20.6 + 20.6i)37-s − 23.3i·39-s + ⋯
L(s)  = 1  + (0.408 + 0.408i)3-s + (−0.0786 + 0.0786i)7-s + 0.333i·9-s + 0.140·11-s + (−0.734 − 0.734i)13-s + (−0.655 + 0.655i)17-s − 0.668i·19-s − 0.0642·21-s + (−0.928 − 0.928i)23-s + (−0.136 + 0.136i)27-s − 1.51i·29-s + 1.43·31-s + (0.0575 + 0.0575i)33-s + (−0.558 + 0.558i)37-s − 0.599i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.229 + 0.973i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.229 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1200\)    =    \(2^{4} \cdot 3 \cdot 5^{2}\)
Sign: $-0.229 + 0.973i$
Analytic conductor: \(32.6976\)
Root analytic conductor: \(5.71818\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1200} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1200,\ (\ :1),\ -0.229 + 0.973i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.023289020\)
\(L(\frac12)\) \(\approx\) \(1.023289020\)
\(L(2)\) 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 \( 1 + (-1.22 - 1.22i)T \)
5 \( 1 \)
good7 \( 1 + (0.550 - 0.550i)T - 49iT^{2} \)
11 \( 1 - 1.55T + 121T^{2} \)
13 \( 1 + (9.55 + 9.55i)T + 169iT^{2} \)
17 \( 1 + (11.1 - 11.1i)T - 289iT^{2} \)
19 \( 1 + 12.6iT - 361T^{2} \)
23 \( 1 + (21.3 + 21.3i)T + 529iT^{2} \)
29 \( 1 + 44.0iT - 841T^{2} \)
31 \( 1 - 44.4T + 961T^{2} \)
37 \( 1 + (20.6 - 20.6i)T - 1.36e3iT^{2} \)
41 \( 1 + 48.2T + 1.68e3T^{2} \)
43 \( 1 + (36.2 + 36.2i)T + 1.84e3iT^{2} \)
47 \( 1 + (-42.5 + 42.5i)T - 2.20e3iT^{2} \)
53 \( 1 + (-54.4 - 54.4i)T + 2.80e3iT^{2} \)
59 \( 1 + 47.4iT - 3.48e3T^{2} \)
61 \( 1 + 59.8T + 3.72e3T^{2} \)
67 \( 1 + (-81.2 + 81.2i)T - 4.48e3iT^{2} \)
71 \( 1 + 87.5T + 5.04e3T^{2} \)
73 \( 1 + (75.9 + 75.9i)T + 5.32e3iT^{2} \)
79 \( 1 + 97.3iT - 6.24e3T^{2} \)
83 \( 1 + (-41.0 - 41.0i)T + 6.88e3iT^{2} \)
89 \( 1 - 52.2iT - 7.92e3T^{2} \)
97 \( 1 + (-37 + 37i)T - 9.40e3iT^{2} \)
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

−9.324793339701862276342244540337, −8.474692083397428812349145751460, −7.902482917303736510197044692601, −6.80974478801260236758729488396, −5.98542277567771656410914349209, −4.86276939573952778361793416613, −4.15028956717892905626598178127, −2.97813784024939668658947059633, −2.08558694775451632283805947882, −0.27381233776609975097632334105, 1.42424388260694442919702039601, 2.47385943338585087728238331087, 3.58204783091550739987964334013, 4.58333179945591549956134926097, 5.60269217299573722245644836075, 6.73288979876455073823156135383, 7.20115798026626550305371442543, 8.201686624439131912561406779767, 8.926175010325594130507221953254, 9.747606040394435347205900801481

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