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Properties

Label 2-2760-2760.2189-c0-0-5
Degree $2$
Conductor $2760$
Sign $0.117 + 0.993i$
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  + (−0.415 − 0.909i)2-s + (0.959 − 0.281i)3-s + (−0.654 + 0.755i)4-s + (0.841 + 0.540i)5-s + (−0.654 − 0.755i)6-s + (0.959 + 0.281i)8-s + (0.841 − 0.540i)9-s + (0.142 − 0.989i)10-s + (0.698 − 1.53i)11-s + (−0.415 + 0.909i)12-s + (−0.0405 + 0.281i)13-s + (0.959 + 0.281i)15-s + (−0.142 − 0.989i)16-s + (−1.25 − 1.45i)17-s + (−0.841 − 0.540i)18-s + ⋯
L(s)  = 1  + (−0.415 − 0.909i)2-s + (0.959 − 0.281i)3-s + (−0.654 + 0.755i)4-s + (0.841 + 0.540i)5-s + (−0.654 − 0.755i)6-s + (0.959 + 0.281i)8-s + (0.841 − 0.540i)9-s + (0.142 − 0.989i)10-s + (0.698 − 1.53i)11-s + (−0.415 + 0.909i)12-s + (−0.0405 + 0.281i)13-s + (0.959 + 0.281i)15-s + (−0.142 − 0.989i)16-s + (−1.25 − 1.45i)17-s + (−0.841 − 0.540i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.117 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.117 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2760\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.117 + 0.993i$
Analytic conductor: \(1.37741\)
Root analytic conductor: \(1.17363\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2760} (2189, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2760,\ (\ :0),\ 0.117 + 0.993i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.552824448\)
\(L(\frac12)\) \(\approx\) \(1.552824448\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.415 + 0.909i)T \)
3 \( 1 + (-0.959 + 0.281i)T \)
5 \( 1 + (-0.841 - 0.540i)T \)
23 \( 1 + (0.841 - 0.540i)T \)
good7 \( 1 + (0.959 - 0.281i)T^{2} \)
11 \( 1 + (-0.698 + 1.53i)T + (-0.654 - 0.755i)T^{2} \)
13 \( 1 + (0.0405 - 0.281i)T + (-0.959 - 0.281i)T^{2} \)
17 \( 1 + (1.25 + 1.45i)T + (-0.142 + 0.989i)T^{2} \)
19 \( 1 + (0.142 + 0.989i)T^{2} \)
29 \( 1 + (0.544 + 0.627i)T + (-0.142 + 0.989i)T^{2} \)
31 \( 1 + (-1.25 - 0.368i)T + (0.841 + 0.540i)T^{2} \)
37 \( 1 + (0.698 - 0.449i)T + (0.415 - 0.909i)T^{2} \)
41 \( 1 + (-0.415 - 0.909i)T^{2} \)
43 \( 1 + (0.273 - 0.0801i)T + (0.841 - 0.540i)T^{2} \)
47 \( 1 - 1.91T + T^{2} \)
53 \( 1 + (0.959 - 0.281i)T^{2} \)
59 \( 1 + (0.239 - 1.66i)T + (-0.959 - 0.281i)T^{2} \)
61 \( 1 + (-0.841 - 0.540i)T^{2} \)
67 \( 1 + (-0.544 - 1.19i)T + (-0.654 + 0.755i)T^{2} \)
71 \( 1 + (0.654 - 0.755i)T^{2} \)
73 \( 1 + (0.142 + 0.989i)T^{2} \)
79 \( 1 + (-0.186 + 1.29i)T + (-0.959 - 0.281i)T^{2} \)
83 \( 1 + (-0.415 + 0.909i)T^{2} \)
89 \( 1 + (-0.841 + 0.540i)T^{2} \)
97 \( 1 + (-0.415 - 0.909i)T^{2} \)
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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

−8.946711805095702894503327148675, −8.419506779501359315096230694680, −7.42437614410976399274893948219, −6.73769047567761261758242690495, −5.86175508537995222096896668697, −4.58399141132890051107753770444, −3.67354311475738211740215483269, −2.87463611222072231895260593509, −2.25232870367774046214846292936, −1.14633714397061525429001056669, 1.64204093949850442464532075036, 2.18164970410789197370957802190, 3.97423482428652784262084124541, 4.46578943057171868704746999153, 5.29161515784403433260339759905, 6.39932749363271989601421170745, 6.82905249050333932989598531579, 7.87314960802606839394129419764, 8.454051704072332560312880666265, 9.103997161279701367236598440693

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