↑

Properties

Label 2-2880-15.14-c0-0-1
Degree $2$
Conductor $2880$
Sign $0.985 - 0.169i$
Analytic cond. $1.43730$
Root an. cond. $1.19887$
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  + (−0.707 − 0.707i)5-s + 2i·13-s + 1.41·17-s + 1.00i·25-s − 1.41i·29-s + 1.41i·41-s + 49-s + 1.41·53-s + 2·61-s + (1.41 − 1.41i)65-s − 2i·73-s + (−1.00 − 1.00i)85-s − 1.41i·89-s + 2i·97-s + 1.41i·101-s + ⋯
L(s)  = 1  + (−0.707 − 0.707i)5-s + 2i·13-s + 1.41·17-s + 1.00i·25-s − 1.41i·29-s + 1.41i·41-s + 49-s + 1.41·53-s + 2·61-s + (1.41 − 1.41i)65-s − 2i·73-s + (−1.00 − 1.00i)85-s − 1.41i·89-s + 2i·97-s + 1.41i·101-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 - 0.169i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 - 0.169i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2880\)    =    \(2^{6} \cdot 3^{2} \cdot 5\)
Sign: $0.985 - 0.169i$
Analytic conductor: \(1.43730\)
Root analytic conductor: \(1.19887\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2880} (449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2880,\ (\ :0),\ 0.985 - 0.169i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.101176601\)
\(L(\frac12)\) \(\approx\) \(1.101176601\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 + (0.707 + 0.707i)T \)
good7 \( 1 - T^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 - 2iT - T^{2} \)
17 \( 1 - 1.41T + T^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 + 1.41iT - T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 - 1.41iT - T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 - 1.41T + T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - 2T + T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + 2iT - T^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + 1.41iT - T^{2} \)
97 \( 1 - 2iT - T^{2} \)
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

−9.003848376129615276659647393927, −8.192232870570653191455840508797, −7.57126604523528563850222856368, −6.78095828297287195451104132497, −5.90901560541997943630413307574, −4.98162742693724104626452718196, −4.22235109297430412652992120081, −3.60859834472351057555068622190, −2.26607244498138164603204743230, −1.11214025496158152223036339484, 0.887174675626570215053767738059, 2.57431080428751018525059167751, 3.32068229020994839574456775130, 3.95496008821509788022553851932, 5.35832663100933243542670358400, 5.62925751551261521691760571943, 6.91219895612642795582472555625, 7.40658985949181730711033165897, 8.142910897516606585289083711307, 8.694780707829462832213492250921

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