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Properties

Label 2-1440-20.19-c2-0-14
Degree $2$
Conductor $1440$
Sign $0.248 - 0.968i$
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  + (−4.30 + 2.54i)5-s − 3.84·7-s + 6.19i·11-s − 16.1i·13-s + 5.20i·17-s − 36.2i·19-s + 22.0·23-s + (12.0 − 21.9i)25-s + 20.0·29-s + 26.4i·31-s + (16.5 − 9.80i)35-s + 69.3i·37-s − 11.6·41-s + 25.8·43-s − 66.1·47-s + ⋯
L(s)  = 1  + (−0.860 + 0.509i)5-s − 0.549·7-s + 0.562i·11-s − 1.23i·13-s + 0.306i·17-s − 1.90i·19-s + 0.958·23-s + (0.480 − 0.876i)25-s + 0.690·29-s + 0.852i·31-s + (0.473 − 0.280i)35-s + 1.87i·37-s − 0.283·41-s + 0.601·43-s − 1.40·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.248 - 0.968i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.248 - 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1440\)    =    \(2^{5} \cdot 3^{2} \cdot 5\)
Sign: $0.248 - 0.968i$
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: \((2,\ 1440,\ (\ :1),\ 0.248 - 0.968i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.077763384\)
\(L(\frac12)\) \(\approx\) \(1.077763384\)
\(L(2)\) 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 \( 1 + (4.30 - 2.54i)T \)
good7 \( 1 + 3.84T + 49T^{2} \)
11 \( 1 - 6.19iT - 121T^{2} \)
13 \( 1 + 16.1iT - 169T^{2} \)
17 \( 1 - 5.20iT - 289T^{2} \)
19 \( 1 + 36.2iT - 361T^{2} \)
23 \( 1 - 22.0T + 529T^{2} \)
29 \( 1 - 20.0T + 841T^{2} \)
31 \( 1 - 26.4iT - 961T^{2} \)
37 \( 1 - 69.3iT - 1.36e3T^{2} \)
41 \( 1 + 11.6T + 1.68e3T^{2} \)
43 \( 1 - 25.8T + 1.84e3T^{2} \)
47 \( 1 + 66.1T + 2.20e3T^{2} \)
53 \( 1 - 39.5iT - 2.80e3T^{2} \)
59 \( 1 - 27.7iT - 3.48e3T^{2} \)
61 \( 1 + 54.1T + 3.72e3T^{2} \)
67 \( 1 - 107.T + 4.48e3T^{2} \)
71 \( 1 - 70.7iT - 5.04e3T^{2} \)
73 \( 1 - 37.4iT - 5.32e3T^{2} \)
79 \( 1 - 97.6iT - 6.24e3T^{2} \)
83 \( 1 - 126.T + 6.88e3T^{2} \)
89 \( 1 + 133.T + 7.92e3T^{2} \)
97 \( 1 - 6.40iT - 9.40e3T^{2} \)
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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

−9.575882821062174337323498506823, −8.587980058746058149919979552865, −7.927291918207594129609953305213, −6.91645140324301657836607134166, −6.60077047980607282326487135659, −5.19704575553120374215729486695, −4.49252144507779066993001090046, −3.21314856761456704319374503259, −2.77327161555408700395043214421, −0.888607392890041604818243645693, 0.40131174848970460839373599109, 1.76973218245728006255749908572, 3.27319075862349916269714938990, 3.95422248699617371813273157712, 4.88134825672903149749794799326, 5.92667744853348377589721704084, 6.74698653441336345711497384435, 7.65673215803430998771769312574, 8.355667055607175442965588271824, 9.178182801730670679940077409101

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