↑

Properties

Label 2-720-80.59-c0-0-1
Degree $2$
Conductor $720$
Sign $0.382 + 0.923i$
Analytic cond. $0.359326$
Root an. cond. $0.599438$
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)2-s + 1.00i·4-s + (0.707 − 0.707i)5-s + (0.707 − 0.707i)8-s − 1.00·10-s − 1.00·16-s − 1.41i·17-s + (1 + i)19-s + (0.707 + 0.707i)20-s + 1.41i·23-s − 1.00i·25-s − 2i·31-s + (0.707 + 0.707i)32-s + (−1.00 + 1.00i)34-s − 1.41i·38-s + ⋯
L(s)  = 1  + (−0.707 − 0.707i)2-s + 1.00i·4-s + (0.707 − 0.707i)5-s + (0.707 − 0.707i)8-s − 1.00·10-s − 1.00·16-s − 1.41i·17-s + (1 + i)19-s + (0.707 + 0.707i)20-s + 1.41i·23-s − 1.00i·25-s − 2i·31-s + (0.707 + 0.707i)32-s + (−1.00 + 1.00i)34-s − 1.41i·38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.382 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.382 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(720\)    =    \(2^{4} \cdot 3^{2} \cdot 5\)
Sign: $0.382 + 0.923i$
Analytic conductor: \(0.359326\)
Root analytic conductor: \(0.599438\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{720} (379, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 720,\ (\ :0),\ 0.382 + 0.923i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7649421831\)
\(L(\frac12)\) \(\approx\) \(0.7649421831\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−10.16082418165504980075734466990, −9.607022466584688324558734475436, −9.085113752990866780665547868187, −7.963615193229931851982675290965, −7.32067924072005349420657053653, −5.94177936700939655715355141630, −4.96641370577638982848358959965, −3.74322955803488103635970183538, −2.50138346045873189197283043621, −1.24946640468036301038987573849, 1.65089928933564993302833260922, 3.01702673398197661037316773425, 4.69447148435520395844182353373, 5.69949412272248130018087159535, 6.55672195386414276608856355551, 7.13931642202132727536844916255, 8.280599784901348970192876446868, 8.993184408357590596372335303342, 9.927025037500139583340724238653, 10.56172363104129373175838486318

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