Properties

Label 2-1440-60.59-c1-0-14
Degree $2$
Conductor $1440$
Sign $0.998 - 0.0547i$
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  + (−0.256 + 2.22i)5-s + 3.50·7-s + 1.92·11-s − 5.50i·13-s + 4.44·17-s − 7.00i·19-s + 1.10i·23-s + (−4.86 − 1.14i)25-s − 5.47i·29-s + 8.28i·31-s + (−0.900 + 7.78i)35-s − 0.778i·37-s − 2.44i·41-s + 9.55·43-s + 11.7i·47-s + ⋯
L(s)  = 1  + (−0.114 + 0.993i)5-s + 1.32·7-s + 0.581·11-s − 1.52i·13-s + 1.07·17-s − 1.60i·19-s + 0.229i·23-s + (−0.973 − 0.228i)25-s − 1.01i·29-s + 1.48i·31-s + (−0.152 + 1.31i)35-s − 0.127i·37-s − 0.381i·41-s + 1.45·43-s + 1.70i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0547i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 - 0.0547i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1440\)    =    \(2^{5} \cdot 3^{2} \cdot 5\)
Sign: $0.998 - 0.0547i$
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: \((2,\ 1440,\ (\ :1/2),\ 0.998 - 0.0547i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.052617733\)
\(L(\frac12)\) \(\approx\) \(2.052617733\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.256 - 2.22i)T \)
good7 \( 1 - 3.50T + 7T^{2} \)
11 \( 1 - 1.92T + 11T^{2} \)
13 \( 1 + 5.50iT - 13T^{2} \)
17 \( 1 - 4.44T + 17T^{2} \)
19 \( 1 + 7.00iT - 19T^{2} \)
23 \( 1 - 1.10iT - 23T^{2} \)
29 \( 1 + 5.47iT - 29T^{2} \)
31 \( 1 - 8.28iT - 31T^{2} \)
37 \( 1 + 0.778iT - 37T^{2} \)
41 \( 1 + 2.44iT - 41T^{2} \)
43 \( 1 - 9.55T + 43T^{2} \)
47 \( 1 - 11.7iT - 47T^{2} \)
53 \( 1 - 11.5T + 53T^{2} \)
59 \( 1 + 9.78T + 59T^{2} \)
61 \( 1 + 3.45T + 61T^{2} \)
67 \( 1 + 5.45T + 67T^{2} \)
71 \( 1 - 4.25T + 71T^{2} \)
73 \( 1 + 7.27iT - 73T^{2} \)
79 \( 1 - 2.82iT - 79T^{2} \)
83 \( 1 - 4.25iT - 83T^{2} \)
89 \( 1 - 0.386iT - 89T^{2} \)
97 \( 1 + 9.29iT - 97T^{2} \)
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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

−9.577292204081108477337372057335, −8.646034723140887516271510327895, −7.70588525255044730084307072921, −7.39934749863478987297508026339, −6.22805602060828749808691137394, −5.40068183590715197473533088834, −4.50333051494677495010837151911, −3.34752654960888040922538328254, −2.49898467695342993638960433727, −1.04387983910098404773703748460, 1.24989195828657484596674129926, 1.94971130945076969315863364004, 3.82171685788763515256557945315, 4.38326498501211271021601536375, 5.31341869306392045709954139306, 6.07336260438682478044592854739, 7.32015499040260345607658066836, 7.988804094208287739976449698882, 8.708774077165203001274183601176, 9.365366334841796305084902621999

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