↑

Properties

Label 2-2240-280.27-c0-0-7
Degree $2$
Conductor $2240$
Sign $-0.584 + 0.811i$
Analytic cond. $1.11790$
Root an. cond. $1.05731$
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  + (−0.541 − 0.541i)3-s + (−0.382 − 0.923i)5-s + (0.707 − 0.707i)7-s − 0.414i·9-s + (1.30 − 1.30i)13-s + (−0.292 + 0.707i)15-s − 0.765·19-s − 0.765·21-s + (1 + i)23-s + (−0.707 + 0.707i)25-s + (−0.765 + 0.765i)27-s + (−0.923 − 0.382i)35-s − 1.41·39-s + (−0.382 + 0.158i)45-s − 1.00i·49-s + ⋯
L(s)  = 1  + (−0.541 − 0.541i)3-s + (−0.382 − 0.923i)5-s + (0.707 − 0.707i)7-s − 0.414i·9-s + (1.30 − 1.30i)13-s + (−0.292 + 0.707i)15-s − 0.765·19-s − 0.765·21-s + (1 + i)23-s + (−0.707 + 0.707i)25-s + (−0.765 + 0.765i)27-s + (−0.923 − 0.382i)35-s − 1.41·39-s + (−0.382 + 0.158i)45-s − 1.00i·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.584 + 0.811i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.584 + 0.811i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2240\)    =    \(2^{6} \cdot 5 \cdot 7\)
Sign: $-0.584 + 0.811i$
Analytic conductor: \(1.11790\)
Root analytic conductor: \(1.05731\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2240} (1567, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2240,\ (\ :0),\ -0.584 + 0.811i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9797398493\)
\(L(\frac12)\) \(\approx\) \(0.9797398493\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (0.382 + 0.923i)T \)
7 \( 1 + (-0.707 + 0.707i)T \)
good3 \( 1 + (0.541 + 0.541i)T + iT^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 + (-1.30 + 1.30i)T - iT^{2} \)
17 \( 1 - iT^{2} \)
19 \( 1 + 0.765T + T^{2} \)
23 \( 1 + (-1 - i)T + iT^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 - iT^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - iT^{2} \)
47 \( 1 + iT^{2} \)
53 \( 1 + iT^{2} \)
59 \( 1 + 1.84T + T^{2} \)
61 \( 1 - 1.84T + T^{2} \)
67 \( 1 + iT^{2} \)
71 \( 1 + 1.41iT - T^{2} \)
73 \( 1 + iT^{2} \)
79 \( 1 + 1.41T + T^{2} \)
83 \( 1 + (-1.30 - 1.30i)T + iT^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 - iT^{2} \)
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

−8.779423640630354632589985787579, −8.128010907355153271159846009895, −7.51221147934694446435909132681, −6.61999183259913124970336349973, −5.72764236195663737223033822911, −5.10614180137880024021326167390, −4.07711252683919474035243975596, −3.34308667657747940811261525034, −1.52829456893358315916389536216, −0.802833112913258038881940146836, 1.79794803264381314795398195389, 2.78302142300204964124869829367, 4.04139210044133113795485208161, 4.56850834774323643349910970737, 5.58713870116918395278798255061, 6.35997903150612424290405440979, 7.01174912241860025032133117063, 8.114791017128816134014549490412, 8.638925187679573940385539015990, 9.479895814916373306939503516026

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