↑

Properties

Label 2-2880-16.5-c1-0-9
Degree $2$
Conductor $2880$
Sign $-0.727 - 0.686i$
Analytic cond. $22.9969$
Root an. cond. $4.79550$
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  + (−0.707 − 0.707i)5-s + 4.92i·7-s + (2.45 + 2.45i)11-s + (−2.93 + 2.93i)13-s + 5.77·17-s + (0.984 − 0.984i)19-s − 0.539i·23-s + 1.00i·25-s + (−6.81 + 6.81i)29-s + 2.63·31-s + (3.48 − 3.48i)35-s + (−6.00 − 6.00i)37-s − 5.17i·41-s + (0.180 + 0.180i)43-s + 5.57·47-s + ⋯
L(s)  = 1  + (−0.316 − 0.316i)5-s + 1.86i·7-s + (0.741 + 0.741i)11-s + (−0.812 + 0.812i)13-s + 1.40·17-s + (0.225 − 0.225i)19-s − 0.112i·23-s + 0.200i·25-s + (−1.26 + 1.26i)29-s + 0.472·31-s + (0.589 − 0.589i)35-s + (−0.987 − 0.987i)37-s − 0.808i·41-s + (0.0274 + 0.0274i)43-s + 0.813·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.727 - 0.686i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.727 - 0.686i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2880\)    =    \(2^{6} \cdot 3^{2} \cdot 5\)
Sign: $-0.727 - 0.686i$
Analytic conductor: \(22.9969\)
Root analytic conductor: \(4.79550\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2880} (2161, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2880,\ (\ :1/2),\ -0.727 - 0.686i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.274324213\)
\(L(\frac12)\) \(\approx\) \(1.274324213\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 + (0.707 + 0.707i)T \)
good7 \( 1 - 4.92iT - 7T^{2} \)
11 \( 1 + (-2.45 - 2.45i)T + 11iT^{2} \)
13 \( 1 + (2.93 - 2.93i)T - 13iT^{2} \)
17 \( 1 - 5.77T + 17T^{2} \)
19 \( 1 + (-0.984 + 0.984i)T - 19iT^{2} \)
23 \( 1 + 0.539iT - 23T^{2} \)
29 \( 1 + (6.81 - 6.81i)T - 29iT^{2} \)
31 \( 1 - 2.63T + 31T^{2} \)
37 \( 1 + (6.00 + 6.00i)T + 37iT^{2} \)
41 \( 1 + 5.17iT - 41T^{2} \)
43 \( 1 + (-0.180 - 0.180i)T + 43iT^{2} \)
47 \( 1 - 5.57T + 47T^{2} \)
53 \( 1 + (-0.146 - 0.146i)T + 53iT^{2} \)
59 \( 1 + (-3.13 - 3.13i)T + 59iT^{2} \)
61 \( 1 + (1.87 - 1.87i)T - 61iT^{2} \)
67 \( 1 + (8.02 - 8.02i)T - 67iT^{2} \)
71 \( 1 + 7.40iT - 71T^{2} \)
73 \( 1 - 11.1iT - 73T^{2} \)
79 \( 1 + 7.71T + 79T^{2} \)
83 \( 1 + (1.62 - 1.62i)T - 83iT^{2} \)
89 \( 1 + 9.54iT - 89T^{2} \)
97 \( 1 - 4.39T + 97T^{2} \)
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

−9.112531147479031584301165853300, −8.517382058155570726383280328814, −7.44845882548102358664907866869, −6.94471800515644045994481290614, −5.74891243597898078905421870624, −5.35629919812558134327668076183, −4.44332122489743441610412310310, −3.42472892309262498583256113448, −2.39327565312772162008705478862, −1.55180114345356362927164493813, 0.42418162323354457685149339377, 1.38132212619906128786407550562, 3.06225422076624007174292800763, 3.63586835846947041939742475590, 4.38504518662463517448902254768, 5.40792892004823903687638246829, 6.29990156155883427624997893178, 7.13672233339556986406777078510, 7.73059047679572027688860713711, 8.147238292122495272820171642792

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