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Properties

Label 2-1440-5.4-c1-0-14
Degree $2$
Conductor $1440$
Sign $0.774 + 0.632i$
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  + (−1.41 + 1.73i)5-s − 2.82i·7-s − 4.89·11-s + 4.89i·13-s − 3.46i·17-s + 6.92·19-s − 4i·23-s + (−0.999 − 4.89i)25-s + 8.48·29-s + 6.92·31-s + (4.89 + 4.00i)35-s − 4.89i·37-s − 5.65·41-s − 11.3i·43-s + 4i·47-s + ⋯
L(s)  = 1  + (−0.632 + 0.774i)5-s − 1.06i·7-s − 1.47·11-s + 1.35i·13-s − 0.840i·17-s + 1.58·19-s − 0.834i·23-s + (−0.199 − 0.979i)25-s + 1.57·29-s + 1.24·31-s + (0.828 + 0.676i)35-s − 0.805i·37-s − 0.883·41-s − 1.72i·43-s + 0.583i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.774 + 0.632i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.774 + 0.632i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1440\)    =    \(2^{5} \cdot 3^{2} \cdot 5\)
Sign: $0.774 + 0.632i$
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: \((2,\ 1440,\ (\ :1/2),\ 0.774 + 0.632i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.230804244\)
\(L(\frac12)\) \(\approx\) \(1.230804244\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.41 - 1.73i)T \)
good7 \( 1 + 2.82iT - 7T^{2} \)
11 \( 1 + 4.89T + 11T^{2} \)
13 \( 1 - 4.89iT - 13T^{2} \)
17 \( 1 + 3.46iT - 17T^{2} \)
19 \( 1 - 6.92T + 19T^{2} \)
23 \( 1 + 4iT - 23T^{2} \)
29 \( 1 - 8.48T + 29T^{2} \)
31 \( 1 - 6.92T + 31T^{2} \)
37 \( 1 + 4.89iT - 37T^{2} \)
41 \( 1 + 5.65T + 41T^{2} \)
43 \( 1 + 11.3iT - 43T^{2} \)
47 \( 1 - 4iT - 47T^{2} \)
53 \( 1 - 3.46iT - 53T^{2} \)
59 \( 1 - 4.89T + 59T^{2} \)
61 \( 1 + 6T + 61T^{2} \)
67 \( 1 + 5.65iT - 67T^{2} \)
71 \( 1 - 9.79T + 71T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 - 6.92T + 79T^{2} \)
83 \( 1 + 16iT - 83T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 - 9.79iT - 97T^{2} \)
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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

−9.616720497759029784844867384641, −8.491541527157070356657306526072, −7.62889575811188101998331666727, −7.14703977158602591133418735600, −6.43502391573128965809123158827, −5.06635177922668515358240308913, −4.37027509543722738856199325097, −3.30772906145806578654120745264, −2.44615266102313414727694626372, −0.61710910129298858356457792409, 1.03727686841702577926665278090, 2.67753335213580441856715098047, 3.38465351393997329561772466976, 4.92623937438994587096466717060, 5.25311216590510760011827317587, 6.13433949214940589423952028083, 7.50710423423217914278777977778, 8.217592572752129456792195990013, 8.453026838915657132272134895495, 9.726763171973623825941782945922

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