Properties

Label 2-1440-9.4-c1-0-39
Degree $2$
Conductor $1440$
Sign $0.311 + 0.950i$
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  + (1.72 − 0.123i)3-s + (−0.5 + 0.866i)5-s + (−1.53 − 2.66i)7-s + (2.96 − 0.426i)9-s + (−1.97 − 3.41i)11-s + (−0.396 + 0.686i)13-s + (−0.756 + 1.55i)15-s + 4.70·17-s + 1.32·19-s + (−2.98 − 4.40i)21-s + (0.536 − 0.929i)23-s + (−0.499 − 0.866i)25-s + (5.07 − 1.10i)27-s + (−4.39 − 7.61i)29-s + (−0.855 + 1.48i)31-s + ⋯
L(s)  = 1  + (0.997 − 0.0712i)3-s + (−0.223 + 0.387i)5-s + (−0.580 − 1.00i)7-s + (0.989 − 0.142i)9-s + (−0.594 − 1.02i)11-s + (−0.109 + 0.190i)13-s + (−0.195 + 0.402i)15-s + 1.14·17-s + 0.304·19-s + (−0.651 − 0.961i)21-s + (0.111 − 0.193i)23-s + (−0.0999 − 0.173i)25-s + (0.977 − 0.212i)27-s + (−0.816 − 1.41i)29-s + (−0.153 + 0.266i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.311 + 0.950i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.311 + 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1440\)    =    \(2^{5} \cdot 3^{2} \cdot 5\)
Sign: $0.311 + 0.950i$
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: \((2,\ 1440,\ (\ :1/2),\ 0.311 + 0.950i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.992853384\)
\(L(\frac12)\) \(\approx\) \(1.992853384\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.72 + 0.123i)T \)
5 \( 1 + (0.5 - 0.866i)T \)
good7 \( 1 + (1.53 + 2.66i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.97 + 3.41i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.396 - 0.686i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 4.70T + 17T^{2} \)
19 \( 1 - 1.32T + 19T^{2} \)
23 \( 1 + (-0.536 + 0.929i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (4.39 + 7.61i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (0.855 - 1.48i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 8.11T + 37T^{2} \)
41 \( 1 + (0.103 - 0.179i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (5.35 + 9.26i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (4.62 + 8.00i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 12.7T + 53T^{2} \)
59 \( 1 + (-5.97 + 10.3i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.98 - 6.89i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.00644 - 0.0111i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 9.44T + 71T^{2} \)
73 \( 1 - 7.14T + 73T^{2} \)
79 \( 1 + (-3.60 - 6.24i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-5.70 - 9.87i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 1.67T + 89T^{2} \)
97 \( 1 + (-7.12 - 12.3i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.562943677085505844703753489038, −8.337624201376308601794703448674, −7.85795273933926002568649874265, −7.10754645847310492491810208871, −6.30026777059592574636011071726, −5.12876793214162590040766109347, −3.76294224483042834057693159109, −3.46050748368360978046935509178, −2.33550084736856216653767254085, −0.72371914460939197506606916172, 1.57034495215221298719239382236, 2.73292629101800490502573239138, 3.42460646933280532445801364084, 4.65981381439847855227882514867, 5.39336696867743550937448967036, 6.50988977697720012643170637372, 7.66186277183221794511102777460, 7.923853586594204827139280289877, 9.065556270459208809675439340512, 9.527979399303189192037995440757

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