Properties

Label 2-1440-60.59-c1-0-21
Degree $2$
Conductor $1440$
Sign $-0.408 + 0.912i$
Analytic cond. $11.4984$
Root an. cond. $3.39093$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.24 − 1.85i)5-s − 0.864·7-s − 3.90·11-s + 1.13i·13-s + 3.71·17-s − 1.72i·19-s − 9.03i·23-s + (−1.89 − 4.62i)25-s − 1.26i·29-s + 3.25i·31-s + (−1.07 + 1.60i)35-s − 6.38i·37-s − 6.39i·41-s − 4.77·43-s − 4.59i·47-s + ⋯
L(s)  = 1  + (0.557 − 0.830i)5-s − 0.326·7-s − 1.17·11-s + 0.314i·13-s + 0.900·17-s − 0.396i·19-s − 1.88i·23-s + (−0.379 − 0.925i)25-s − 0.235i·29-s + 0.584i·31-s + (−0.182 + 0.271i)35-s − 1.05i·37-s − 0.998i·41-s − 0.728·43-s − 0.670i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.408 + 0.912i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.408 + 0.912i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1440\)    =    \(2^{5} \cdot 3^{2} \cdot 5\)
Sign: $-0.408 + 0.912i$
Analytic conductor: \(11.4984\)
Root analytic conductor: \(3.39093\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1440} (1439, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1440,\ (\ :1/2),\ -0.408 + 0.912i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.229868910\)
\(L(\frac12)\) \(\approx\) \(1.229868910\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.24 + 1.85i)T \)
good7 \( 1 + 0.864T + 7T^{2} \)
11 \( 1 + 3.90T + 11T^{2} \)
13 \( 1 - 1.13iT - 13T^{2} \)
17 \( 1 - 3.71T + 17T^{2} \)
19 \( 1 + 1.72iT - 19T^{2} \)
23 \( 1 + 9.03iT - 23T^{2} \)
29 \( 1 + 1.26iT - 29T^{2} \)
31 \( 1 - 3.25iT - 31T^{2} \)
37 \( 1 + 6.38iT - 37T^{2} \)
41 \( 1 + 6.39iT - 41T^{2} \)
43 \( 1 + 4.77T + 43T^{2} \)
47 \( 1 + 4.59iT - 47T^{2} \)
53 \( 1 - 8.98T + 53T^{2} \)
59 \( 1 + 8.50T + 59T^{2} \)
61 \( 1 + 9.04T + 61T^{2} \)
67 \( 1 + 11.0T + 67T^{2} \)
71 \( 1 - 8.10T + 71T^{2} \)
73 \( 1 - 4.47iT - 73T^{2} \)
79 \( 1 - 14.2iT - 79T^{2} \)
83 \( 1 + 8.10iT - 83T^{2} \)
89 \( 1 + 3.56iT - 89T^{2} \)
97 \( 1 + 10.9iT - 97T^{2} \)
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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

−9.186244443913742430164199284487, −8.578022292464826655456503874831, −7.77449083065489220318397245586, −6.79263195841962821696336337054, −5.85429531620061968779470692298, −5.13500346067501952696163927222, −4.33008732928604118556354699172, −3.00187496837760268079805651586, −1.99974473450207240968897367451, −0.47606375874807381956501904621, 1.59906520014057908728074675087, 2.90273839826163103142414246679, 3.44163350883987852265966905810, 4.96078896964535959262933039743, 5.73514892953801224732436988982, 6.40496605310224534700232946540, 7.60522969384160935620551379191, 7.82740413723794183755326600182, 9.188037512747428726543840481302, 9.950010256054014432694477370401

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