Properties

Label 2-1440-5.4-c1-0-12
Degree $2$
Conductor $1440$
Sign $0.447 - 0.894i$
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 + i)5-s + 2i·7-s + 6·11-s + 2i·13-s + 6i·17-s − 4·19-s − 8i·23-s + (3 + 4i)25-s − 8·31-s + (−2 + 4i)35-s + 2i·37-s + 6·41-s − 4i·43-s + 4i·47-s + 3·49-s + ⋯
L(s)  = 1  + (0.894 + 0.447i)5-s + 0.755i·7-s + 1.80·11-s + 0.554i·13-s + 1.45i·17-s − 0.917·19-s − 1.66i·23-s + (0.600 + 0.800i)25-s − 1.43·31-s + (−0.338 + 0.676i)35-s + 0.328i·37-s + 0.937·41-s − 0.609i·43-s + 0.583i·47-s + 0.428·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1440\)    =    \(2^{5} \cdot 3^{2} \cdot 5\)
Sign: $0.447 - 0.894i$
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: \((2,\ 1440,\ (\ :1/2),\ 0.447 - 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.096746509\)
\(L(\frac12)\) \(\approx\) \(2.096746509\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2 - i)T \)
good7 \( 1 - 2iT - 7T^{2} \)
11 \( 1 - 6T + 11T^{2} \)
13 \( 1 - 2iT - 13T^{2} \)
17 \( 1 - 6iT - 17T^{2} \)
19 \( 1 + 4T + 19T^{2} \)
23 \( 1 + 8iT - 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 + 8T + 31T^{2} \)
37 \( 1 - 2iT - 37T^{2} \)
41 \( 1 - 6T + 41T^{2} \)
43 \( 1 + 4iT - 43T^{2} \)
47 \( 1 - 4iT - 47T^{2} \)
53 \( 1 - 6iT - 53T^{2} \)
59 \( 1 + 6T + 59T^{2} \)
61 \( 1 + 6T + 61T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 - 4T + 71T^{2} \)
73 \( 1 + 12iT - 73T^{2} \)
79 \( 1 - 8T + 79T^{2} \)
83 \( 1 - 12iT - 83T^{2} \)
89 \( 1 - 14T + 89T^{2} \)
97 \( 1 - 8iT - 97T^{2} \)
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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

−9.356617050526758854595170777932, −9.107199685103969193972183334531, −8.291088575153106100783681739681, −6.94959191496572905790491693111, −6.28663596806027908069130909675, −5.89977247781588452141840122716, −4.52740208341722857855538290370, −3.69671675833016336473403765613, −2.35336805806330563720610955749, −1.57524137386085790187701855969, 0.905665913566149566844231977803, 1.94363061949545825485439880304, 3.39455638567554239278118331441, 4.27630864557566518018787456996, 5.26259218881425705491593562626, 6.10493581300060518518083163955, 6.94645429673339141052476489839, 7.64935600880662294895007811941, 8.922545014685414204835721066245, 9.317303426941891077866547460954

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