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Properties

Label 2-2760-2760.1019-c0-0-1
Degree $2$
Conductor $2760$
Sign $0.739 + 0.673i$
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  + (−0.959 + 0.281i)2-s + (−0.841 + 0.540i)3-s + (0.841 − 0.540i)4-s + (−0.415 − 0.909i)5-s + (0.654 − 0.755i)6-s + (−0.654 + 0.755i)8-s + (0.415 − 0.909i)9-s + (0.654 + 0.755i)10-s + (−0.415 + 0.909i)12-s + (0.841 + 0.540i)15-s + (0.415 − 0.909i)16-s + (1.95 + 0.281i)17-s + (−0.142 + 0.989i)18-s + (−0.557 + 0.0801i)19-s + (−0.841 − 0.540i)20-s + ⋯
L(s)  = 1  + (−0.959 + 0.281i)2-s + (−0.841 + 0.540i)3-s + (0.841 − 0.540i)4-s + (−0.415 − 0.909i)5-s + (0.654 − 0.755i)6-s + (−0.654 + 0.755i)8-s + (0.415 − 0.909i)9-s + (0.654 + 0.755i)10-s + (−0.415 + 0.909i)12-s + (0.841 + 0.540i)15-s + (0.415 − 0.909i)16-s + (1.95 + 0.281i)17-s + (−0.142 + 0.989i)18-s + (−0.557 + 0.0801i)19-s + (−0.841 − 0.540i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.739 + 0.673i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.739 + 0.673i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2760\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.739 + 0.673i$
Analytic conductor: \(1.37741\)
Root analytic conductor: \(1.17363\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2760} (1019, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2760,\ (\ :0),\ 0.739 + 0.673i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4896093792\)
\(L(\frac12)\) \(\approx\) \(0.4896093792\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.959 - 0.281i)T \)
3 \( 1 + (0.841 - 0.540i)T \)
5 \( 1 + (0.415 + 0.909i)T \)
23 \( 1 + (0.654 + 0.755i)T \)
good7 \( 1 + (-0.841 + 0.540i)T^{2} \)
11 \( 1 + (-0.142 + 0.989i)T^{2} \)
13 \( 1 + (0.841 + 0.540i)T^{2} \)
17 \( 1 + (-1.95 - 0.281i)T + (0.959 + 0.281i)T^{2} \)
19 \( 1 + (0.557 - 0.0801i)T + (0.959 - 0.281i)T^{2} \)
29 \( 1 + (-0.959 - 0.281i)T^{2} \)
31 \( 1 + (0.304 - 0.474i)T + (-0.415 - 0.909i)T^{2} \)
37 \( 1 + (0.654 + 0.755i)T^{2} \)
41 \( 1 + (0.654 - 0.755i)T^{2} \)
43 \( 1 + (0.415 - 0.909i)T^{2} \)
47 \( 1 + 1.51iT - T^{2} \)
53 \( 1 + (-1.25 + 0.368i)T + (0.841 - 0.540i)T^{2} \)
59 \( 1 + (-0.841 - 0.540i)T^{2} \)
61 \( 1 + (1.10 + 0.708i)T + (0.415 + 0.909i)T^{2} \)
67 \( 1 + (-0.142 - 0.989i)T^{2} \)
71 \( 1 + (-0.142 - 0.989i)T^{2} \)
73 \( 1 + (0.959 - 0.281i)T^{2} \)
79 \( 1 + (-0.797 - 0.234i)T + (0.841 + 0.540i)T^{2} \)
83 \( 1 + (0.983 + 0.449i)T + (0.654 + 0.755i)T^{2} \)
89 \( 1 + (0.415 - 0.909i)T^{2} \)
97 \( 1 + (-0.654 + 0.755i)T^{2} \)
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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

−8.842067388539740470299919848729, −8.304801293308710071064479302933, −7.50365013738323364746415910364, −6.68181785597823761124913098198, −5.70551649587507349786311919656, −5.35129947311498498953310200879, −4.28253334754968819999890459293, −3.35473324126935374257786155687, −1.71920083284105525277150700336, −0.58845314569380257384551186915, 1.08051152895786267759375813021, 2.26123668187750800889030949288, 3.21787769204383313414631696555, 4.18369400908562595498051341464, 5.63008660003847589907847676175, 6.14559726501141517177199626883, 7.05295651393527094350655234881, 7.66633453552936606069683030975, 7.971973938524431898564745437862, 9.191115974458007599893891715589

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