↑

Properties

Label 2-1050-5.2-c2-0-8
Degree $2$
Conductor $1050$
Sign $-0.973 + 0.229i$
Analytic cond. $28.6104$
Root an. cond. $5.34887$
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 + i)2-s + (−1.22 + 1.22i)3-s + 2i·4-s − 2.44·6-s + (1.87 + 1.87i)7-s + (−2 + 2i)8-s − 2.99i·9-s + 15.1·11-s + (−2.44 − 2.44i)12-s + (−14.6 + 14.6i)13-s + 3.74i·14-s − 4·16-s + (−15.9 − 15.9i)17-s + (2.99 − 2.99i)18-s + 2.05i·19-s + ⋯
L(s)  = 1  + (0.5 + 0.5i)2-s + (−0.408 + 0.408i)3-s + 0.5i·4-s − 0.408·6-s + (0.267 + 0.267i)7-s + (−0.250 + 0.250i)8-s − 0.333i·9-s + 1.37·11-s + (−0.204 − 0.204i)12-s + (−1.12 + 1.12i)13-s + 0.267i·14-s − 0.250·16-s + (−0.940 − 0.940i)17-s + (0.166 − 0.166i)18-s + 0.107i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.973 + 0.229i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.973 + 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $-0.973 + 0.229i$
Analytic conductor: \(28.6104\)
Root analytic conductor: \(5.34887\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1050} (757, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1050,\ (\ :1),\ -0.973 + 0.229i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.227434423\)
\(L(\frac12)\) \(\approx\) \(1.227434423\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1 - i)T \)
3 \( 1 + (1.22 - 1.22i)T \)
5 \( 1 \)
7 \( 1 + (-1.87 - 1.87i)T \)
good11 \( 1 - 15.1T + 121T^{2} \)
13 \( 1 + (14.6 - 14.6i)T - 169iT^{2} \)
17 \( 1 + (15.9 + 15.9i)T + 289iT^{2} \)
19 \( 1 - 2.05iT - 361T^{2} \)
23 \( 1 + (19.4 - 19.4i)T - 529iT^{2} \)
29 \( 1 - 33.1iT - 841T^{2} \)
31 \( 1 - 27.8T + 961T^{2} \)
37 \( 1 + (-30.4 - 30.4i)T + 1.36e3iT^{2} \)
41 \( 1 + 26.6T + 1.68e3T^{2} \)
43 \( 1 + (7.89 - 7.89i)T - 1.84e3iT^{2} \)
47 \( 1 + (33.8 + 33.8i)T + 2.20e3iT^{2} \)
53 \( 1 + (5.60 - 5.60i)T - 2.80e3iT^{2} \)
59 \( 1 - 5.80iT - 3.48e3T^{2} \)
61 \( 1 + 98.2T + 3.72e3T^{2} \)
67 \( 1 + (51.6 + 51.6i)T + 4.48e3iT^{2} \)
71 \( 1 + 120.T + 5.04e3T^{2} \)
73 \( 1 + (81.9 - 81.9i)T - 5.32e3iT^{2} \)
79 \( 1 - 33.0iT - 6.24e3T^{2} \)
83 \( 1 + (-97.0 + 97.0i)T - 6.88e3iT^{2} \)
89 \( 1 + 34.2iT - 7.92e3T^{2} \)
97 \( 1 + (104. + 104. i)T + 9.40e3iT^{2} \)
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

−9.968715010553471192400980157892, −9.301553421988619801721012948596, −8.629978659849324459084785174553, −7.36276596392172595417090284686, −6.72089174384376571817282444070, −5.95719344290921280820433556123, −4.74111612136572631034616776855, −4.42618940315881296492707104822, −3.15366939597831312323415999307, −1.73409307643421083987323211537, 0.32497903873989306262805490429, 1.63550570972518732119858815375, 2.72011658239810703128438761178, 4.09980931476189519785069865420, 4.69353848770237335892279420121, 5.95550318090173408508212206670, 6.46889545905387260197488682891, 7.54717405361150133300211421347, 8.396908401337030191612524688804, 9.481063530373667713700805632218

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