↑

Properties

Label 2-1800-360.347-c0-0-2
Degree $2$
Conductor $1800$
Sign $0.116 - 0.993i$
Analytic cond. $0.898317$
Root an. cond. $0.947795$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.965 + 0.258i)2-s + (−0.258 + 0.965i)3-s + (0.866 + 0.499i)4-s + (−0.499 + 0.866i)6-s + (0.707 + 0.707i)8-s + (−0.866 − 0.499i)9-s + (1.5 − 0.866i)11-s + (−0.707 + 0.707i)12-s + (0.500 + 0.866i)16-s + (−0.707 + 0.707i)17-s + (−0.707 − 0.707i)18-s + i·19-s + (1.67 − 0.448i)22-s + (−0.866 + 0.500i)24-s + (0.707 − 0.707i)27-s + ⋯
L(s)  = 1  + (0.965 + 0.258i)2-s + (−0.258 + 0.965i)3-s + (0.866 + 0.499i)4-s + (−0.499 + 0.866i)6-s + (0.707 + 0.707i)8-s + (−0.866 − 0.499i)9-s + (1.5 − 0.866i)11-s + (−0.707 + 0.707i)12-s + (0.500 + 0.866i)16-s + (−0.707 + 0.707i)17-s + (−0.707 − 0.707i)18-s + i·19-s + (1.67 − 0.448i)22-s + (−0.866 + 0.500i)24-s + (0.707 − 0.707i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.116 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.116 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1800\)    =    \(2^{3} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.116 - 0.993i$
Analytic conductor: \(0.898317\)
Root analytic conductor: \(0.947795\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1800} (707, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1800,\ (\ :0),\ 0.116 - 0.993i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.968956410\)
\(L(\frac12)\) \(\approx\) \(1.968956410\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.965 - 0.258i)T \)
3 \( 1 + (0.258 - 0.965i)T \)
5 \( 1 \)
good7 \( 1 + (0.866 + 0.5i)T^{2} \)
11 \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \)
13 \( 1 + (0.866 - 0.5i)T^{2} \)
17 \( 1 + (0.707 - 0.707i)T - iT^{2} \)
19 \( 1 - iT - T^{2} \)
23 \( 1 + (-0.866 + 0.5i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 - iT^{2} \)
41 \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \)
43 \( 1 + (1.67 + 0.448i)T + (0.866 + 0.5i)T^{2} \)
47 \( 1 + (-0.866 - 0.5i)T^{2} \)
53 \( 1 - iT^{2} \)
59 \( 1 + (-0.866 + 1.5i)T + (-0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (-1.67 + 0.448i)T + (0.866 - 0.5i)T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (1.22 - 1.22i)T - iT^{2} \)
79 \( 1 + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + (-0.517 + 1.93i)T + (-0.866 - 0.5i)T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + (0.448 - 1.67i)T + (-0.866 - 0.5i)T^{2} \)
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

−9.778480808026806109247240786824, −8.643908240725626686218500653805, −8.316937788372193961742678233962, −6.83164014416569120514138836681, −6.32625651852690267667933217092, −5.55914196292768435859230051155, −4.70841884075712064905390133552, −3.68637544252553756347205513559, −3.49121543569572155018620683639, −1.84310965810612597520566317995, 1.31007127493659229095360362619, 2.24288356610607951842524322984, 3.28331328081662677597770206235, 4.48518277291294117730704224536, 5.09005332031294500503907237940, 6.25027319000922646611968505345, 6.80913056421222819378205074131, 7.22025309703834028010800342997, 8.412345208350741381674378660192, 9.349725960141697994451509073479

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