Properties

Label 2-1440-5.4-c1-0-19
Degree $2$
Conductor $1440$
Sign $i$
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  − 2.23·5-s + 0.763i·7-s − 5.70i·23-s + 5.00·25-s − 6·29-s − 1.70i·35-s + 4.47·41-s − 11.2i·43-s − 13.7i·47-s + 6.41·49-s − 13.4·61-s − 8.18i·67-s − 17.7i·83-s + 6·89-s + 18·101-s + ⋯
L(s)  = 1  − 0.999·5-s + 0.288i·7-s − 1.19i·23-s + 1.00·25-s − 1.11·29-s − 0.288i·35-s + 0.698·41-s − 1.71i·43-s − 1.99i·47-s + 0.916·49-s − 1.71·61-s − 0.999i·67-s − 1.94i·83-s + 0.635·89-s + 1.79·101-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1440\)    =    \(2^{5} \cdot 3^{2} \cdot 5\)
Sign: $i$
Analytic conductor: \(11.4984\)
Root analytic conductor: \(3.39093\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1440} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1440,\ (\ :1/2),\ i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8844060155\)
\(L(\frac12)\) \(\approx\) \(0.8844060155\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.23T \)
good7 \( 1 - 0.763iT - 7T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 - 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 + 5.70iT - 23T^{2} \)
29 \( 1 + 6T + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 - 37T^{2} \)
41 \( 1 - 4.47T + 41T^{2} \)
43 \( 1 + 11.2iT - 43T^{2} \)
47 \( 1 + 13.7iT - 47T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 + 13.4T + 61T^{2} \)
67 \( 1 + 8.18iT - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + 17.7iT - 83T^{2} \)
89 \( 1 - 6T + 89T^{2} \)
97 \( 1 - 97T^{2} \)
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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

−9.078370359371748802396301680386, −8.616405583855486674573862430536, −7.67272406988954743806120204033, −7.05451846298634568720811072803, −6.05787805785935546198358658898, −5.06810304912545001160237796950, −4.16114513807988639728471297203, −3.31906518039096453568612848383, −2.12072966994198831819881376486, −0.39156806227144678853467505921, 1.23629850438442708890282226677, 2.82538235659231648077356102088, 3.78889472190645049311628637546, 4.52923907614230466791568109167, 5.57816554126748423400448726305, 6.55526190724563738863186979149, 7.62886311933258916854510515325, 7.79346287836696513965330525748, 9.003148411219530185346330034822, 9.586164339750300985145667727421

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