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Properties

Label 2-1440-9.4-c1-0-23
Degree $2$
Conductor $1440$
Sign $0.766 - 0.642i$
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  + (1.5 + 0.866i)3-s + (−0.5 + 0.866i)5-s + (−1.5 − 2.59i)7-s + (1.5 + 2.59i)9-s + (2 + 3.46i)11-s + (2 − 3.46i)13-s + (−1.5 + 0.866i)15-s + 2·17-s + 2·19-s − 5.19i·21-s + (3.5 − 6.06i)23-s + (−0.499 − 0.866i)25-s + 5.19i·27-s + (4.5 + 7.79i)29-s + (−3 + 5.19i)31-s + ⋯
L(s)  = 1  + (0.866 + 0.499i)3-s + (−0.223 + 0.387i)5-s + (−0.566 − 0.981i)7-s + (0.5 + 0.866i)9-s + (0.603 + 1.04i)11-s + (0.554 − 0.960i)13-s + (−0.387 + 0.223i)15-s + 0.485·17-s + 0.458·19-s − 1.13i·21-s + (0.729 − 1.26i)23-s + (−0.0999 − 0.173i)25-s + 0.999i·27-s + (0.835 + 1.44i)29-s + (−0.538 + 0.933i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1440\)    =    \(2^{5} \cdot 3^{2} \cdot 5\)
Sign: $0.766 - 0.642i$
Analytic conductor: \(11.4984\)
Root analytic conductor: \(3.39093\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1440} (481, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1440,\ (\ :1/2),\ 0.766 - 0.642i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.277879680\)
\(L(\frac12)\) \(\approx\) \(2.277879680\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.5 - 0.866i)T \)
5 \( 1 + (0.5 - 0.866i)T \)
good7 \( 1 + (1.5 + 2.59i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2 - 3.46i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2 + 3.46i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 2T + 17T^{2} \)
19 \( 1 - 2T + 19T^{2} \)
23 \( 1 + (-3.5 + 6.06i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-4.5 - 7.79i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (3 - 5.19i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 2T + 37T^{2} \)
41 \( 1 + (4.5 - 7.79i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (2 + 3.46i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-1.5 - 2.59i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 6T + 53T^{2} \)
59 \( 1 + (-3 + 5.19i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6.5 - 11.2i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.5 - 2.59i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 8T + 73T^{2} \)
79 \( 1 + (5 + 8.66i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (3.5 + 6.06i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - T + 89T^{2} \)
97 \( 1 + (-7 - 12.1i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.711676928177112821308923450659, −8.820509884844381138709402753007, −8.077001323888815233106077750829, −7.09647897740824745320782206073, −6.77301527900561703522936967243, −5.25545604249761292761246106319, −4.34169950456451886517474604303, −3.49435726826145050355321227370, −2.84595659644318422784676185891, −1.24681582128021905837638373273, 1.02074679397857580838969339398, 2.27032257874278009754569112778, 3.36095979141425679052665092176, 3.99332336757437231258778983069, 5.48529482331468642891240993658, 6.20910172218835029345130070436, 7.03247170236136401176869984337, 8.034347358198727872612474359148, 8.684205340604873195326364701413, 9.286516392215275375626881020447

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