↑

Properties

Label 2-1200-5.2-c2-0-20
Degree $2$
Conductor $1200$
Sign $0.973 - 0.229i$
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 + (6.44 + 6.44i)7-s − 2.99i·9-s + 11.7·11-s + (2.44 − 2.44i)13-s + (0.898 + 0.898i)17-s − 33.5i·19-s − 15.7·21-s + (21.7 − 21.7i)23-s + (3.67 + 3.67i)27-s + 3.59i·29-s + 25.1·31-s + (−14.4 + 14.4i)33-s + (9.14 + 9.14i)37-s + 5.99i·39-s + ⋯
L(s)  = 1  + (−0.408 + 0.408i)3-s + (0.921 + 0.921i)7-s − 0.333i·9-s + 1.07·11-s + (0.188 − 0.188i)13-s + (0.0528 + 0.0528i)17-s − 1.76i·19-s − 0.752·21-s + (0.947 − 0.947i)23-s + (0.136 + 0.136i)27-s + 0.123i·29-s + 0.812·31-s + (−0.437 + 0.437i)33-s + (0.247 + 0.247i)37-s + 0.153i·39-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1200\)    =    \(2^{4} \cdot 3 \cdot 5^{2}\)
Sign: $0.973 - 0.229i$
Analytic conductor: \(32.6976\)
Root analytic conductor: \(5.71818\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1200} (1057, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1200,\ (\ :1),\ 0.973 - 0.229i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.093679097\)
\(L(\frac12)\) \(\approx\) \(2.093679097\)
\(L(2)\) 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 \( 1 + (-6.44 - 6.44i)T + 49iT^{2} \)
11 \( 1 - 11.7T + 121T^{2} \)
13 \( 1 + (-2.44 + 2.44i)T - 169iT^{2} \)
17 \( 1 + (-0.898 - 0.898i)T + 289iT^{2} \)
19 \( 1 + 33.5iT - 361T^{2} \)
23 \( 1 + (-21.7 + 21.7i)T - 529iT^{2} \)
29 \( 1 - 3.59iT - 841T^{2} \)
31 \( 1 - 25.1T + 961T^{2} \)
37 \( 1 + (-9.14 - 9.14i)T + 1.36e3iT^{2} \)
41 \( 1 + 60.7T + 1.68e3T^{2} \)
43 \( 1 + (22.2 - 22.2i)T - 1.84e3iT^{2} \)
47 \( 1 + (-4.49 - 4.49i)T + 2.20e3iT^{2} \)
53 \( 1 + (-52.0 + 52.0i)T - 2.80e3iT^{2} \)
59 \( 1 + 86.9iT - 3.48e3T^{2} \)
61 \( 1 + 108.T + 3.72e3T^{2} \)
67 \( 1 + (-77.8 - 77.8i)T + 4.48e3iT^{2} \)
71 \( 1 - 66.7T + 5.04e3T^{2} \)
73 \( 1 + (-78.6 + 78.6i)T - 5.32e3iT^{2} \)
79 \( 1 - 66iT - 6.24e3T^{2} \)
83 \( 1 + (-84.0 + 84.0i)T - 6.88e3iT^{2} \)
89 \( 1 - 9.59iT - 7.92e3T^{2} \)
97 \( 1 + (-53.0 - 53.0i)T + 9.40e3iT^{2} \)
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

−9.447158819923032818641675064886, −8.822776353648822098847529881371, −8.201184404616997789309929410502, −6.86931395402635682793978640102, −6.31189077259142773881089939127, −5.04332043608061405143473986385, −4.75838571989406891630853245673, −3.41684928128431408872540643889, −2.23974224438614818917391488029, −0.865957850751069624457649288029, 1.04274050703057721056799296918, 1.74192862740613911815301044592, 3.50395843452495333426592964898, 4.31469959694048762757159208761, 5.30598812845279994443659990887, 6.25574172121299180618507659986, 7.08634883410941589104616208284, 7.79538161208443017810404324176, 8.567446057860362620280856088482, 9.600439578666085164674159332237

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