↑

Properties

Label 2-1800-24.11-c1-0-63
Degree $2$
Conductor $1800$
Sign $0.577 + 0.816i$
Analytic cond. $14.3730$
Root an. cond. $3.79118$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 1.41·2-s + 2.00·4-s − 5.16i·7-s + 2.82·8-s + 3.05i·11-s − 0.837i·13-s − 7.30i·14-s + 4.00·16-s + 6.32·19-s + 4.32i·22-s − 4.47·23-s − 1.18i·26-s − 10.3i·28-s + 5.65·32-s − 11.1i·37-s + 8.94·38-s + ⋯
L(s)  = 1  + 1.00·2-s + 1.00·4-s − 1.95i·7-s + 1.00·8-s + 0.921i·11-s − 0.232i·13-s − 1.95i·14-s + 1.00·16-s + 1.45·19-s + 0.921i·22-s − 0.932·23-s − 0.232i·26-s − 1.95i·28-s + 1.00·32-s − 1.83i·37-s + 1.45·38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1800\)    =    \(2^{3} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.577 + 0.816i$
Analytic conductor: \(14.3730\)
Root analytic conductor: \(3.79118\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1800} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1800,\ (\ :1/2),\ 0.577 + 0.816i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.478118084\)
\(L(\frac12)\) \(\approx\) \(3.478118084\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 1.41T \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + 5.16iT - 7T^{2} \)
11 \( 1 - 3.05iT - 11T^{2} \)
13 \( 1 + 0.837iT - 13T^{2} \)
17 \( 1 - 17T^{2} \)
19 \( 1 - 6.32T + 19T^{2} \)
23 \( 1 + 4.47T + 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 + 11.1iT - 37T^{2} \)
41 \( 1 + 10.3iT - 41T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 - 2.82T + 47T^{2} \)
53 \( 1 - 5.65T + 53T^{2} \)
59 \( 1 - 5.42iT - 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 + 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 73T^{2} \)
79 \( 1 - 79T^{2} \)
83 \( 1 - 83T^{2} \)
89 \( 1 - 18.8iT - 89T^{2} \)
97 \( 1 + 97T^{2} \)
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

−9.384089733172585891818916255276, −7.892665154517199710166820173441, −7.33993877177040417721805180719, −6.95488782047875033872842152047, −5.80617689869238197980760728931, −4.94739790694104715792766241976, −4.03070294074259257715342503512, −3.61636991370735424336812758395, −2.22622683775010673410677295826, −0.975880346395970499719312029130, 1.61066314243921419217924791097, 2.76004086341226496253128191456, 3.28631764000586852284197760797, 4.61812234233835667818559886127, 5.46132358404450566645939972307, 5.94027327666820655928432868950, 6.67385016181546764863891346720, 7.892858420023505944727777989068, 8.463189319824399548593935149028, 9.413754437560428083803884040091

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