↑

Properties

Label 2-2800-112.67-c0-0-2
Degree $2$
Conductor $2800$
Sign $0.981 + 0.193i$
Analytic cond. $1.39738$
Root an. cond. $1.18210$
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  + (0.866 − 0.5i)2-s + (0.366 + 1.36i)3-s + (0.499 − 0.866i)4-s + (1 + 0.999i)6-s + (−0.5 − 0.866i)7-s − 0.999i·8-s + (−0.866 + 0.5i)9-s + (1.36 + 0.366i)12-s + (−0.866 − 0.499i)14-s + (−0.5 − 0.866i)16-s + (0.5 − 0.866i)17-s + (−0.499 + 0.866i)18-s + (1.36 + 0.366i)19-s + (0.999 − i)21-s + (0.5 + 0.866i)23-s + (1.36 − 0.366i)24-s + ⋯
L(s)  = 1  + (0.866 − 0.5i)2-s + (0.366 + 1.36i)3-s + (0.499 − 0.866i)4-s + (1 + 0.999i)6-s + (−0.5 − 0.866i)7-s − 0.999i·8-s + (−0.866 + 0.5i)9-s + (1.36 + 0.366i)12-s + (−0.866 − 0.499i)14-s + (−0.5 − 0.866i)16-s + (0.5 − 0.866i)17-s + (−0.499 + 0.866i)18-s + (1.36 + 0.366i)19-s + (0.999 − i)21-s + (0.5 + 0.866i)23-s + (1.36 − 0.366i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.981 + 0.193i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.981 + 0.193i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2800\)    =    \(2^{4} \cdot 5^{2} \cdot 7\)
Sign: $0.981 + 0.193i$
Analytic conductor: \(1.39738\)
Root analytic conductor: \(1.18210\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2800} (851, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2800,\ (\ :0),\ 0.981 + 0.193i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.284716814\)
\(L(\frac12)\) \(\approx\) \(2.284716814\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 + 0.5i)T \)
5 \( 1 \)
7 \( 1 + (0.5 + 0.866i)T \)
good3 \( 1 + (-0.366 - 1.36i)T + (-0.866 + 0.5i)T^{2} \)
11 \( 1 + (0.866 - 0.5i)T^{2} \)
13 \( 1 + iT^{2} \)
17 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
19 \( 1 + (-1.36 - 0.366i)T + (0.866 + 0.5i)T^{2} \)
23 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
29 \( 1 + iT^{2} \)
31 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
37 \( 1 + (0.866 + 0.5i)T^{2} \)
41 \( 1 - iT - T^{2} \)
43 \( 1 + (1 + i)T + iT^{2} \)
47 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
53 \( 1 + (0.366 + 1.36i)T + (-0.866 + 0.5i)T^{2} \)
59 \( 1 + (-1.36 + 0.366i)T + (0.866 - 0.5i)T^{2} \)
61 \( 1 + (-0.366 + 1.36i)T + (-0.866 - 0.5i)T^{2} \)
67 \( 1 + (-0.866 + 0.5i)T^{2} \)
71 \( 1 + T + T^{2} \)
73 \( 1 + (0.5 + 0.866i)T^{2} \)
79 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
83 \( 1 + (1 - i)T - iT^{2} \)
89 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
97 \( 1 + T + T^{2} \)
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

−9.536946692645381735225876285193, −8.315859437614178918456724807671, −7.26591777235857107223134287889, −6.62834119918275521839960425692, −5.38840720386748163102454104760, −4.99326905612140571524922219553, −4.08203201792224667073055256519, −3.35133251327227795450573096667, −2.96970646721092132507391567044, −1.26275387093626049571147617988, 1.49457723455521136508580858335, 2.64926496857458915312635466767, 3.08693300396154314713042078986, 4.32715018981503439640670378393, 5.44034403849361548245854255080, 5.99702693578327147439459490011, 6.78654687385261230228680369504, 7.25894135412494149290567403872, 8.190165135684341432598776753627, 8.527333444750704572092343405443

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