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Properties

Label 2-1440-9.4-c1-0-35
Degree $2$
Conductor $1440$
Sign $0.283 + 0.958i$
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  + (0.0981 + 1.72i)3-s + (0.5 − 0.866i)5-s + (−0.0981 − 0.170i)7-s + (−2.98 + 0.339i)9-s + (−2.64 − 4.58i)11-s + (−1.28 + 2.22i)13-s + (1.54 + 0.779i)15-s − 5.22·17-s + 6.89·19-s + (0.284 − 0.186i)21-s + (3.66 − 6.35i)23-s + (−0.499 − 0.866i)25-s + (−0.879 − 5.12i)27-s + (−2.39 − 4.15i)29-s + (1.12 − 1.94i)31-s + ⋯
L(s)  = 1  + (0.0566 + 0.998i)3-s + (0.223 − 0.387i)5-s + (−0.0371 − 0.0642i)7-s + (−0.993 + 0.113i)9-s + (−0.797 − 1.38i)11-s + (−0.356 + 0.617i)13-s + (0.399 + 0.201i)15-s − 1.26·17-s + 1.58·19-s + (0.0620 − 0.0406i)21-s + (0.764 − 1.32i)23-s + (−0.0999 − 0.173i)25-s + (−0.169 − 0.985i)27-s + (−0.445 − 0.770i)29-s + (0.201 − 0.348i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.283 + 0.958i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.283 + 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1440\)    =    \(2^{5} \cdot 3^{2} \cdot 5\)
Sign: $0.283 + 0.958i$
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: \((2,\ 1440,\ (\ :1/2),\ 0.283 + 0.958i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.001996609\)
\(L(\frac12)\) \(\approx\) \(1.001996609\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.0981 - 1.72i)T \)
5 \( 1 + (-0.5 + 0.866i)T \)
good7 \( 1 + (0.0981 + 0.170i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (2.64 + 4.58i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.28 - 2.22i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 5.22T + 17T^{2} \)
19 \( 1 - 6.89T + 19T^{2} \)
23 \( 1 + (-3.66 + 6.35i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.39 + 4.15i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1.12 + 1.94i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 9.61T + 37T^{2} \)
41 \( 1 + (-2.50 + 4.33i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.44 + 7.70i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-6.44 - 11.1i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 7.53T + 53T^{2} \)
59 \( 1 + (-2.19 + 3.80i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.30 - 5.73i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.74 + 4.75i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 5.49T + 71T^{2} \)
73 \( 1 + 2.57T + 73T^{2} \)
79 \( 1 + (1.19 + 2.07i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (2.38 + 4.13i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 15.5T + 89T^{2} \)
97 \( 1 + (-6.26 - 10.8i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.191605461724470242978564206940, −8.839892148252624713853719026059, −7.994080397264995785301093152906, −6.87088045653507306593501209225, −5.81405229553118040093513449790, −5.15946222449204454989366697028, −4.34014300603629870031538213023, −3.29989570774989090841508825128, −2.36641893999345688074900187013, −0.38771809275831092945674035323, 1.47797100675866047693068205920, 2.50364095661754572613190769984, 3.33843624856744392818279379945, 4.98044543544624791024366344281, 5.48659645893576905078696687048, 6.72644759712625235631703047983, 7.27429993108790938186688372998, 7.78177813682879611509176389870, 8.903400016749102585167417087075, 9.636651673146232271032059736785

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