↑

Properties

Label 2-600-15.14-c2-0-16
Degree $2$
Conductor $600$
Sign $0.752 + 0.659i$
Analytic cond. $16.3488$
Root an. cond. $4.04336$
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  + (−1.13 − 2.77i)3-s + 5.85i·7-s + (−6.42 + 6.29i)9-s − 12.4i·11-s + 11.8i·13-s + 29.2·17-s − 3.19·19-s + (16.2 − 6.64i)21-s + 19.5·23-s + (24.7 + 10.7i)27-s − 30.3i·29-s + 3.57·31-s + (−34.6 + 14.1i)33-s + 42.7i·37-s + (32.9 − 13.4i)39-s + ⋯
L(s)  = 1  + (−0.377 − 0.925i)3-s + 0.836i·7-s + (−0.714 + 0.699i)9-s − 1.13i·11-s + 0.912i·13-s + 1.72·17-s − 0.168·19-s + (0.774 − 0.316i)21-s + 0.849·23-s + (0.917 + 0.396i)27-s − 1.04i·29-s + 0.115·31-s + (−1.04 + 0.428i)33-s + 1.15i·37-s + (0.844 − 0.344i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.752 + 0.659i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.752 + 0.659i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(600\)    =    \(2^{3} \cdot 3 \cdot 5^{2}\)
Sign: $0.752 + 0.659i$
Analytic conductor: \(16.3488\)
Root analytic conductor: \(4.04336\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{600} (449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 600,\ (\ :1),\ 0.752 + 0.659i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.550406816\)
\(L(\frac12)\) \(\approx\) \(1.550406816\)
\(L(2)\) 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 \( 1 + (1.13 + 2.77i)T \)
5 \( 1 \)
good7 \( 1 - 5.85iT - 49T^{2} \)
11 \( 1 + 12.4iT - 121T^{2} \)
13 \( 1 - 11.8iT - 169T^{2} \)
17 \( 1 - 29.2T + 289T^{2} \)
19 \( 1 + 3.19T + 361T^{2} \)
23 \( 1 - 19.5T + 529T^{2} \)
29 \( 1 + 30.3iT - 841T^{2} \)
31 \( 1 - 3.57T + 961T^{2} \)
37 \( 1 - 42.7iT - 1.36e3T^{2} \)
41 \( 1 + 6.39iT - 1.68e3T^{2} \)
43 \( 1 + 62.6iT - 1.84e3T^{2} \)
47 \( 1 - 69.3T + 2.20e3T^{2} \)
53 \( 1 + 57.7T + 2.80e3T^{2} \)
59 \( 1 + 78.9iT - 3.48e3T^{2} \)
61 \( 1 - 68.5T + 3.72e3T^{2} \)
67 \( 1 + 90.5iT - 4.48e3T^{2} \)
71 \( 1 - 26.5iT - 5.04e3T^{2} \)
73 \( 1 - 40.0iT - 5.32e3T^{2} \)
79 \( 1 - 148.T + 6.24e3T^{2} \)
83 \( 1 + 9.36T + 6.88e3T^{2} \)
89 \( 1 + 109. iT - 7.92e3T^{2} \)
97 \( 1 - 161. iT - 9.40e3T^{2} \)
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

−10.52856567225607321202472387637, −9.335148268346378853114203195452, −8.485689767563604741182524839833, −7.74481212350453579359244695305, −6.64596961292720067000516219001, −5.86784561148262990387862704053, −5.11827721976248103207056404082, −3.43235294102368497149168604234, −2.25363718246552387915947057352, −0.863439673916029432879657315799, 0.960852401470601674534261005489, 2.99796000517946691413061990103, 3.99230245536823880061288558999, 4.97110003175812658282021490872, 5.77635461099531088195013216948, 7.06865188735894894153639342281, 7.81632189264652396202267360401, 9.042598197652849253931318706809, 9.934244804703548022648723639248, 10.41771225393165662141328216725

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