Properties

Label 2-2760-2760.1589-c0-0-3
Degree $2$
Conductor $2760$
Sign $0.451 + 0.892i$
Analytic cond. $1.37741$
Root an. cond. $1.17363$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.841 − 0.540i)2-s + (0.142 + 0.989i)3-s + (0.415 + 0.909i)4-s + (−0.959 − 0.281i)5-s + (0.415 − 0.909i)6-s + (0.142 − 0.989i)8-s + (−0.959 + 0.281i)9-s + (0.654 + 0.755i)10-s + (−1.61 + 1.03i)11-s + (−0.841 + 0.540i)12-s + (−0.857 − 0.989i)13-s + (0.142 − 0.989i)15-s + (−0.654 + 0.755i)16-s + (0.118 − 0.258i)17-s + (0.959 + 0.281i)18-s + ⋯
L(s)  = 1  + (−0.841 − 0.540i)2-s + (0.142 + 0.989i)3-s + (0.415 + 0.909i)4-s + (−0.959 − 0.281i)5-s + (0.415 − 0.909i)6-s + (0.142 − 0.989i)8-s + (−0.959 + 0.281i)9-s + (0.654 + 0.755i)10-s + (−1.61 + 1.03i)11-s + (−0.841 + 0.540i)12-s + (−0.857 − 0.989i)13-s + (0.142 − 0.989i)15-s + (−0.654 + 0.755i)16-s + (0.118 − 0.258i)17-s + (0.959 + 0.281i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.451 + 0.892i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.451 + 0.892i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2760\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.451 + 0.892i$
Analytic conductor: \(1.37741\)
Root analytic conductor: \(1.17363\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2760} (1589, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2760,\ (\ :0),\ 0.451 + 0.892i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3985116875\)
\(L(\frac12)\) \(\approx\) \(0.3985116875\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.841 + 0.540i)T \)
3 \( 1 + (-0.142 - 0.989i)T \)
5 \( 1 + (0.959 + 0.281i)T \)
23 \( 1 + (-0.959 + 0.281i)T \)
good7 \( 1 + (0.142 + 0.989i)T^{2} \)
11 \( 1 + (1.61 - 1.03i)T + (0.415 - 0.909i)T^{2} \)
13 \( 1 + (0.857 + 0.989i)T + (-0.142 + 0.989i)T^{2} \)
17 \( 1 + (-0.118 + 0.258i)T + (-0.654 - 0.755i)T^{2} \)
19 \( 1 + (0.654 - 0.755i)T^{2} \)
29 \( 1 + (-0.698 + 1.53i)T + (-0.654 - 0.755i)T^{2} \)
31 \( 1 + (0.118 - 0.822i)T + (-0.959 - 0.281i)T^{2} \)
37 \( 1 + (-1.61 + 0.474i)T + (0.841 - 0.540i)T^{2} \)
41 \( 1 + (-0.841 - 0.540i)T^{2} \)
43 \( 1 + (0.186 + 1.29i)T + (-0.959 + 0.281i)T^{2} \)
47 \( 1 - 0.284T + T^{2} \)
53 \( 1 + (0.142 + 0.989i)T^{2} \)
59 \( 1 + (-1.25 - 1.45i)T + (-0.142 + 0.989i)T^{2} \)
61 \( 1 + (0.959 + 0.281i)T^{2} \)
67 \( 1 + (0.698 + 0.449i)T + (0.415 + 0.909i)T^{2} \)
71 \( 1 + (-0.415 - 0.909i)T^{2} \)
73 \( 1 + (0.654 - 0.755i)T^{2} \)
79 \( 1 + (0.544 + 0.627i)T + (-0.142 + 0.989i)T^{2} \)
83 \( 1 + (-0.841 + 0.540i)T^{2} \)
89 \( 1 + (0.959 - 0.281i)T^{2} \)
97 \( 1 + (-0.841 - 0.540i)T^{2} \)
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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

−8.911640157077623017690315559402, −8.143862059484581157137976021441, −7.70316140831572799021861832963, −7.03678248010521369881107864032, −5.47371510640362838918640443832, −4.77364938336321150672558540761, −4.06273136983760488882099889995, −2.91033535022226539978346873290, −2.50395829607779896669172978532, −0.39170754691701450919055257679, 1.02271367745274596900914871673, 2.48252876905491602501581738897, 3.08278220692420238046345974599, 4.67597325008722637292716975360, 5.49890031713292977255519739682, 6.41529803205179081143868226211, 7.05854503955512849379069440074, 7.73979378196078867980655116713, 8.132207797142967043724790450564, 8.829198587832584975602466226547

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