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Properties

Label 2-1440-60.59-c1-0-9
Degree $2$
Conductor $1440$
Sign $0.948 + 0.318i$
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  + (−2.20 − 0.342i)5-s − 2.64·7-s + 3.00·11-s + 0.640i·13-s − 0.685·17-s + 5.28i·19-s − 2.27i·23-s + (4.76 + 1.51i)25-s − 8.15i·29-s + 2.96i·31-s + (5.83 + 0.905i)35-s − 1.60i·37-s − 7.42i·41-s + 11.2·43-s − 4.19i·47-s + ⋯
L(s)  = 1  + (−0.988 − 0.153i)5-s − 0.997·7-s + 0.906·11-s + 0.177i·13-s − 0.166·17-s + 1.21i·19-s − 0.474i·23-s + (0.952 + 0.303i)25-s − 1.51i·29-s + 0.533i·31-s + (0.986 + 0.152i)35-s − 0.264i·37-s − 1.15i·41-s + 1.71·43-s − 0.612i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.948 + 0.318i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.948 + 0.318i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1440\)    =    \(2^{5} \cdot 3^{2} \cdot 5\)
Sign: $0.948 + 0.318i$
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: \((2,\ 1440,\ (\ :1/2),\ 0.948 + 0.318i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.152839295\)
\(L(\frac12)\) \(\approx\) \(1.152839295\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.20 + 0.342i)T \)
good7 \( 1 + 2.64T + 7T^{2} \)
11 \( 1 - 3.00T + 11T^{2} \)
13 \( 1 - 0.640iT - 13T^{2} \)
17 \( 1 + 0.685T + 17T^{2} \)
19 \( 1 - 5.28iT - 19T^{2} \)
23 \( 1 + 2.27iT - 23T^{2} \)
29 \( 1 + 8.15iT - 29T^{2} \)
31 \( 1 - 2.96iT - 31T^{2} \)
37 \( 1 + 1.60iT - 37T^{2} \)
41 \( 1 + 7.42iT - 41T^{2} \)
43 \( 1 - 11.2T + 43T^{2} \)
47 \( 1 + 4.19iT - 47T^{2} \)
53 \( 1 - 9.60T + 53T^{2} \)
59 \( 1 - 7.20T + 59T^{2} \)
61 \( 1 - 10.4T + 61T^{2} \)
67 \( 1 - 8.49T + 67T^{2} \)
71 \( 1 - 13.1T + 71T^{2} \)
73 \( 1 + 14.2iT - 73T^{2} \)
79 \( 1 - 11.4iT - 79T^{2} \)
83 \( 1 - 13.1iT - 83T^{2} \)
89 \( 1 + 10.2iT - 89T^{2} \)
97 \( 1 - 8.31iT - 97T^{2} \)
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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

−9.451222800842131406827530887041, −8.686113777283695092626910221015, −7.922156424277266742756194411479, −6.99163349167101788440919148714, −6.37111826977304190962440468434, −5.37015926643540173389998802065, −3.93227507008835796855950984574, −3.83201843134876719502437835402, −2.38337444113963610201629463194, −0.69746678738677049751651981768, 0.839302470653202356447040217359, 2.67124296248821395076828981116, 3.55164532760487340166756258216, 4.30830087221508781657623613832, 5.41126439860578103982937435933, 6.62105369870310976923623000881, 6.95771021880307710913849972429, 7.935539402746736956834118989408, 8.895943662293186219134785826418, 9.414213888871423873030324916298

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