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Properties

Label 2-2760-2760.1589-c0-0-2
Degree $2$
Conductor $2760$
Sign $0.947 - 0.320i$
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  + (−0.755 − 0.654i)2-s + (0.989 − 0.142i)3-s + (0.142 + 0.989i)4-s + (−0.281 + 0.959i)5-s + (−0.841 − 0.540i)6-s + (0.540 − 0.841i)8-s + (0.959 − 0.281i)9-s + (0.841 − 0.540i)10-s + (0.281 + 0.959i)12-s + (−0.142 + 0.989i)15-s + (−0.959 + 0.281i)16-s + (−0.755 + 1.65i)17-s + (−0.909 − 0.415i)18-s + (1.37 − 0.627i)19-s + (−0.989 − 0.142i)20-s + ⋯
L(s)  = 1  + (−0.755 − 0.654i)2-s + (0.989 − 0.142i)3-s + (0.142 + 0.989i)4-s + (−0.281 + 0.959i)5-s + (−0.841 − 0.540i)6-s + (0.540 − 0.841i)8-s + (0.959 − 0.281i)9-s + (0.841 − 0.540i)10-s + (0.281 + 0.959i)12-s + (−0.142 + 0.989i)15-s + (−0.959 + 0.281i)16-s + (−0.755 + 1.65i)17-s + (−0.909 − 0.415i)18-s + (1.37 − 0.627i)19-s + (−0.989 − 0.142i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.947 - 0.320i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.947 - 0.320i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2760\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.947 - 0.320i$
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: \((2,\ 2760,\ (\ :0),\ 0.947 - 0.320i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.155706968\)
\(L(\frac12)\) \(\approx\) \(1.155706968\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.755 + 0.654i)T \)
3 \( 1 + (-0.989 + 0.142i)T \)
5 \( 1 + (0.281 - 0.959i)T \)
23 \( 1 + (-0.540 - 0.841i)T \)
good7 \( 1 + (0.142 + 0.989i)T^{2} \)
11 \( 1 + (0.415 - 0.909i)T^{2} \)
13 \( 1 + (-0.142 + 0.989i)T^{2} \)
17 \( 1 + (0.755 - 1.65i)T + (-0.654 - 0.755i)T^{2} \)
19 \( 1 + (-1.37 + 0.627i)T + (0.654 - 0.755i)T^{2} \)
29 \( 1 + (-0.654 - 0.755i)T^{2} \)
31 \( 1 + (0.186 - 1.29i)T + (-0.959 - 0.281i)T^{2} \)
37 \( 1 + (0.841 - 0.540i)T^{2} \)
41 \( 1 + (-0.841 - 0.540i)T^{2} \)
43 \( 1 + (-0.959 + 0.281i)T^{2} \)
47 \( 1 + 1.08T + T^{2} \)
53 \( 1 + (1.27 + 1.10i)T + (0.142 + 0.989i)T^{2} \)
59 \( 1 + (-0.142 + 0.989i)T^{2} \)
61 \( 1 + (-1.07 - 0.153i)T + (0.959 + 0.281i)T^{2} \)
67 \( 1 + (0.415 + 0.909i)T^{2} \)
71 \( 1 + (-0.415 - 0.909i)T^{2} \)
73 \( 1 + (0.654 - 0.755i)T^{2} \)
79 \( 1 + (-1.25 - 1.45i)T + (-0.142 + 0.989i)T^{2} \)
83 \( 1 + (0.0801 + 0.273i)T + (-0.841 + 0.540i)T^{2} \)
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

−9.001374603529011484419282713321, −8.360544523443787628093849018986, −7.70596217359178142346864369339, −7.00687553717746435096772394961, −6.46511652290204364377401447291, −4.89437453834479952920020195466, −3.63060532301289903058706385289, −3.40030934486086521218194194514, −2.37094238805826448339413195070, −1.47685889694227359458338880898, 0.922270917603345922997764293100, 2.10609044586986062374687933185, 3.20175781160177735125275088814, 4.52235747648954249867626274703, 4.93110711103778093483324029324, 5.97773272865695977876592017297, 7.07041736616103721511810031171, 7.62081828226324521234775719467, 8.208675275851849469124039439524, 8.986943058363802941161956833161

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