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Properties

Label 2-1440-20.19-c2-0-29
Degree $2$
Conductor $1440$
Sign $0.997 - 0.0639i$
Analytic cond. $39.2371$
Root an. cond. $6.26395$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (3.75 + 3.30i)5-s − 10.0·7-s − 17.2i·11-s − 4.41i·13-s + 27.0i·17-s + 4.82i·19-s + 15.2·23-s + (3.19 + 24.7i)25-s + 2.38·29-s − 38.0i·31-s + (−37.7 − 33.2i)35-s − 16.5i·37-s + 13.3·41-s + 59.7·43-s + 62.4·47-s + ⋯
L(s)  = 1  + (0.750 + 0.660i)5-s − 1.43·7-s − 1.56i·11-s − 0.339i·13-s + 1.58i·17-s + 0.254i·19-s + 0.663·23-s + (0.127 + 0.991i)25-s + 0.0821·29-s − 1.22i·31-s + (−1.07 − 0.948i)35-s − 0.447i·37-s + 0.324·41-s + 1.39·43-s + 1.32·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0639i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.997 - 0.0639i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1440\)    =    \(2^{5} \cdot 3^{2} \cdot 5\)
Sign: $0.997 - 0.0639i$
Analytic conductor: \(39.2371\)
Root analytic conductor: \(6.26395\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1440} (1279, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1440,\ (\ :1),\ 0.997 - 0.0639i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.811272817\)
\(L(\frac12)\) \(\approx\) \(1.811272817\)
\(L(2)\) 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 \( 1 + (-3.75 - 3.30i)T \)
good7 \( 1 + 10.0T + 49T^{2} \)
11 \( 1 + 17.2iT - 121T^{2} \)
13 \( 1 + 4.41iT - 169T^{2} \)
17 \( 1 - 27.0iT - 289T^{2} \)
19 \( 1 - 4.82iT - 361T^{2} \)
23 \( 1 - 15.2T + 529T^{2} \)
29 \( 1 - 2.38T + 841T^{2} \)
31 \( 1 + 38.0iT - 961T^{2} \)
37 \( 1 + 16.5iT - 1.36e3T^{2} \)
41 \( 1 - 13.3T + 1.68e3T^{2} \)
43 \( 1 - 59.7T + 1.84e3T^{2} \)
47 \( 1 - 62.4T + 2.20e3T^{2} \)
53 \( 1 + 71.5iT - 2.80e3T^{2} \)
59 \( 1 - 68.8iT - 3.48e3T^{2} \)
61 \( 1 - 40.9T + 3.72e3T^{2} \)
67 \( 1 + 51.0T + 4.48e3T^{2} \)
71 \( 1 + 40.4iT - 5.04e3T^{2} \)
73 \( 1 - 35.8iT - 5.32e3T^{2} \)
79 \( 1 - 126. iT - 6.24e3T^{2} \)
83 \( 1 - 75.1T + 6.88e3T^{2} \)
89 \( 1 - 106.T + 7.92e3T^{2} \)
97 \( 1 + 85.4iT - 9.40e3T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.348957793386120893697577565027, −8.727133771355500334666364360135, −7.70029242604205297524926660820, −6.69286017806951612177754641506, −5.93101340276569309972535958891, −5.73554069084334890484256482287, −3.90726543278559055856299699984, −3.25414778571811424028122534908, −2.35230325639795223674513084619, −0.73906482053928158022989786444, 0.78481666367126748524879605045, 2.19335012540467320533160114442, 3.06850321420730588746076914921, 4.43908976097474067706381998128, 5.07050967650796482647224701072, 6.11740318825511891142759707896, 6.92087911181689544477081979286, 7.46656425944291938143501289854, 9.052498255223999931977229310205, 9.220918646702473821265381380988

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