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Properties

Label 2-1440-20.19-c2-0-25
Degree $2$
Conductor $1440$
Sign $0.520 - 0.853i$
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  + (4.85 − 1.17i)5-s + 7.06·7-s − 1.70i·11-s + 19.3i·13-s + 13.0i·17-s + 13.2i·19-s + 5.26·23-s + (22.2 − 11.4i)25-s − 33.0·29-s + 60.5i·31-s + (34.3 − 8.32i)35-s + 46.9i·37-s − 65.3·41-s + 20.2·43-s + 40.0·47-s + ⋯
L(s)  = 1  + (0.971 − 0.235i)5-s + 1.00·7-s − 0.154i·11-s + 1.49i·13-s + 0.765i·17-s + 0.696i·19-s + 0.228·23-s + (0.888 − 0.457i)25-s − 1.13·29-s + 1.95i·31-s + (0.980 − 0.237i)35-s + 1.26i·37-s − 1.59·41-s + 0.470·43-s + 0.852·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.520 - 0.853i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.520 - 0.853i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1440\)    =    \(2^{5} \cdot 3^{2} \cdot 5\)
Sign: $0.520 - 0.853i$
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: \((2,\ 1440,\ (\ :1),\ 0.520 - 0.853i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.558283330\)
\(L(\frac12)\) \(\approx\) \(2.558283330\)
\(L(2)\) 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 \( 1 + (-4.85 + 1.17i)T \)
good7 \( 1 - 7.06T + 49T^{2} \)
11 \( 1 + 1.70iT - 121T^{2} \)
13 \( 1 - 19.3iT - 169T^{2} \)
17 \( 1 - 13.0iT - 289T^{2} \)
19 \( 1 - 13.2iT - 361T^{2} \)
23 \( 1 - 5.26T + 529T^{2} \)
29 \( 1 + 33.0T + 841T^{2} \)
31 \( 1 - 60.5iT - 961T^{2} \)
37 \( 1 - 46.9iT - 1.36e3T^{2} \)
41 \( 1 + 65.3T + 1.68e3T^{2} \)
43 \( 1 - 20.2T + 1.84e3T^{2} \)
47 \( 1 - 40.0T + 2.20e3T^{2} \)
53 \( 1 - 11.2iT - 2.80e3T^{2} \)
59 \( 1 + 39.8iT - 3.48e3T^{2} \)
61 \( 1 - 32.2T + 3.72e3T^{2} \)
67 \( 1 - 2.76T + 4.48e3T^{2} \)
71 \( 1 + 67.6iT - 5.04e3T^{2} \)
73 \( 1 + 81.8iT - 5.32e3T^{2} \)
79 \( 1 + 143. iT - 6.24e3T^{2} \)
83 \( 1 - 126.T + 6.88e3T^{2} \)
89 \( 1 + 11.2T + 7.92e3T^{2} \)
97 \( 1 - 49.5iT - 9.40e3T^{2} \)
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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

−9.342593722513231249569338232337, −8.755039298595969306139050953734, −8.017946548518038476761438150473, −6.90837374195618001871564258140, −6.22142551058578855403536382675, −5.22985536352300510625328187187, −4.59771873828216631739328715863, −3.44167785878641915437834810938, −1.92247631068213789304408000815, −1.48202310657416534727122606477, 0.70893574799531806474612915330, 2.02911784429635771521373700088, 2.84293612693448724234088190755, 4.15828239870950592934089512643, 5.42589577245984044551785148331, 5.50966511410102259369646405799, 6.86234838177081431023623637276, 7.59303529410671975664847072802, 8.384257749145477370373141029216, 9.339809763432334559339092081862

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