Properties

Label 2-2520-2520.293-c0-0-3
Degree $2$
Conductor $2520$
Sign $0.999 - 0.0136i$
Analytic cond. $1.25764$
Root an. cond. $1.12144$
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  + (0.258 − 0.965i)2-s + (0.923 + 0.382i)3-s + (−0.866 − 0.499i)4-s + (0.130 + 0.991i)5-s + (0.608 − 0.793i)6-s + (0.965 + 0.258i)7-s + (−0.707 + 0.707i)8-s + (0.707 + 0.707i)9-s + (0.991 + 0.130i)10-s + (−0.608 − 0.793i)12-s + (−0.739 + 0.198i)13-s + (0.499 − 0.866i)14-s + (−0.258 + 0.965i)15-s + (0.500 + 0.866i)16-s + (0.866 − 0.5i)18-s + 1.58i·19-s + ⋯
L(s)  = 1  + (0.258 − 0.965i)2-s + (0.923 + 0.382i)3-s + (−0.866 − 0.499i)4-s + (0.130 + 0.991i)5-s + (0.608 − 0.793i)6-s + (0.965 + 0.258i)7-s + (−0.707 + 0.707i)8-s + (0.707 + 0.707i)9-s + (0.991 + 0.130i)10-s + (−0.608 − 0.793i)12-s + (−0.739 + 0.198i)13-s + (0.499 − 0.866i)14-s + (−0.258 + 0.965i)15-s + (0.500 + 0.866i)16-s + (0.866 − 0.5i)18-s + 1.58i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0136i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0136i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2520\)    =    \(2^{3} \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.999 - 0.0136i$
Analytic conductor: \(1.25764\)
Root analytic conductor: \(1.12144\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2520} (293, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2520,\ (\ :0),\ 0.999 - 0.0136i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.852219492\)
\(L(\frac12)\) \(\approx\) \(1.852219492\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.258 + 0.965i)T \)
3 \( 1 + (-0.923 - 0.382i)T \)
5 \( 1 + (-0.130 - 0.991i)T \)
7 \( 1 + (-0.965 - 0.258i)T \)
good11 \( 1 + (-0.5 + 0.866i)T^{2} \)
13 \( 1 + (0.739 - 0.198i)T + (0.866 - 0.5i)T^{2} \)
17 \( 1 - iT^{2} \)
19 \( 1 - 1.58iT - T^{2} \)
23 \( 1 + (0.5 + 1.86i)T + (-0.866 + 0.5i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 + iT^{2} \)
41 \( 1 + (-0.5 - 0.866i)T^{2} \)
43 \( 1 + (0.866 + 0.5i)T^{2} \)
47 \( 1 + (0.866 + 0.5i)T^{2} \)
53 \( 1 + iT^{2} \)
59 \( 1 + (-0.923 + 1.60i)T + (-0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.608 + 1.05i)T + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.866 - 0.5i)T^{2} \)
71 \( 1 + 0.517iT - T^{2} \)
73 \( 1 - iT^{2} \)
79 \( 1 + (0.448 - 0.258i)T + (0.5 - 0.866i)T^{2} \)
83 \( 1 + (-1.78 - 0.478i)T + (0.866 + 0.5i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (-0.866 - 0.5i)T^{2} \)
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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

−9.345982457771090718730497448024, −8.220946423956538102604283850832, −8.047635034003965538328140472821, −6.83506877493168459051074466674, −5.79890613541236349964134852231, −4.83445545121273301895724783376, −4.12981512052399918942784620199, −3.28904626276815583210406092636, −2.34281226271393057627217924143, −1.84421288783975281463581377949, 1.11394163894436973670651784146, 2.39481762225651305885779140614, 3.64979120012138503977658641896, 4.50839343050454304091290382873, 5.09723664344136126087228579377, 5.94915916938820277294130598244, 7.20395781312772222890240278809, 7.48600272603008584266797254452, 8.245244390125979768463276818877, 8.909981474495522561365333094234

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