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Properties

Label 2-2496-312.59-c0-0-3
Degree $2$
Conductor $2496$
Sign $-0.233 + 0.972i$
Analytic cond. $1.24566$
Root an. cond. $1.11609$
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  + (0.5 − 0.866i)3-s + (0.133 − 0.5i)7-s + (−0.499 − 0.866i)9-s + (−0.866 + 0.5i)13-s + (0.366 − 1.36i)19-s + (−0.366 − 0.366i)21-s − i·25-s − 0.999·27-s + (1.36 − 1.36i)31-s + (−1.36 + 0.366i)37-s + 0.999i·39-s + (0.866 − 0.5i)43-s + (0.633 + 0.366i)49-s + (−0.999 − i)57-s + (−0.866 + 0.5i)61-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)3-s + (0.133 − 0.5i)7-s + (−0.499 − 0.866i)9-s + (−0.866 + 0.5i)13-s + (0.366 − 1.36i)19-s + (−0.366 − 0.366i)21-s − i·25-s − 0.999·27-s + (1.36 − 1.36i)31-s + (−1.36 + 0.366i)37-s + 0.999i·39-s + (0.866 − 0.5i)43-s + (0.633 + 0.366i)49-s + (−0.999 − i)57-s + (−0.866 + 0.5i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.233 + 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.233 + 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2496\)    =    \(2^{6} \cdot 3 \cdot 13\)
Sign: $-0.233 + 0.972i$
Analytic conductor: \(1.24566\)
Root analytic conductor: \(1.11609\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2496} (2399, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2496,\ (\ :0),\ -0.233 + 0.972i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.287244608\)
\(L(\frac12)\) \(\approx\) \(1.287244608\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.5 + 0.866i)T \)
13 \( 1 + (0.866 - 0.5i)T \)
good5 \( 1 + iT^{2} \)
7 \( 1 + (-0.133 + 0.5i)T + (-0.866 - 0.5i)T^{2} \)
11 \( 1 + (0.866 - 0.5i)T^{2} \)
17 \( 1 + (-0.5 + 0.866i)T^{2} \)
19 \( 1 + (-0.366 + 1.36i)T + (-0.866 - 0.5i)T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 + (-0.5 - 0.866i)T^{2} \)
31 \( 1 + (-1.36 + 1.36i)T - iT^{2} \)
37 \( 1 + (1.36 - 0.366i)T + (0.866 - 0.5i)T^{2} \)
41 \( 1 + (-0.866 + 0.5i)T^{2} \)
43 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
47 \( 1 - iT^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + (-0.866 - 0.5i)T^{2} \)
61 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.5 - 0.133i)T + (0.866 - 0.5i)T^{2} \)
71 \( 1 + (-0.866 - 0.5i)T^{2} \)
73 \( 1 + (-0.366 + 0.366i)T - iT^{2} \)
79 \( 1 - 1.73iT - T^{2} \)
83 \( 1 - iT^{2} \)
89 \( 1 + (0.866 - 0.5i)T^{2} \)
97 \( 1 + (0.5 - 1.86i)T + (-0.866 - 0.5i)T^{2} \)
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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

−8.828114123858920054868142065778, −8.032652344164921161942164488748, −7.31246798381799790775470463195, −6.80970520504917519731923375155, −5.97983267166276215685742683831, −4.84633484350411841906878057583, −4.06767484707079852693662532683, −2.85900411752761296474430016003, −2.18236534610030329694531345954, −0.804647898068817346502806250613, 1.77484794444131334455209268037, 2.89403190274989759179588307267, 3.54871748142952499290944302511, 4.65010498193238132387826834744, 5.28627355015494738689279689123, 6.01270788494567949783661299101, 7.25982427768462984186911939304, 7.906326279459783654165073306257, 8.680191453647401546587958871600, 9.260710298729083020265766884949

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