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Properties

Label 2-2808-104.51-c0-0-7
Degree $2$
Conductor $2808$
Sign $0.707 + 0.707i$
Analytic cond. $1.40137$
Root an. cond. $1.18379$
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  + (0.965 − 0.258i)2-s + (0.866 − 0.499i)4-s + 0.517·5-s + (0.707 − 0.707i)8-s + (0.499 − 0.133i)10-s − 1.41i·11-s + i·13-s + (0.500 − 0.866i)16-s + (0.448 − 0.258i)20-s + (−0.366 − 1.36i)22-s − 0.732·25-s + (0.258 + 0.965i)26-s + (0.258 − 0.965i)32-s + (0.366 − 0.366i)40-s + 1.41i·41-s + ⋯
L(s)  = 1  + (0.965 − 0.258i)2-s + (0.866 − 0.499i)4-s + 0.517·5-s + (0.707 − 0.707i)8-s + (0.499 − 0.133i)10-s − 1.41i·11-s + i·13-s + (0.500 − 0.866i)16-s + (0.448 − 0.258i)20-s + (−0.366 − 1.36i)22-s − 0.732·25-s + (0.258 + 0.965i)26-s + (0.258 − 0.965i)32-s + (0.366 − 0.366i)40-s + 1.41i·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2808 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2808 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2808\)    =    \(2^{3} \cdot 3^{3} \cdot 13\)
Sign: $0.707 + 0.707i$
Analytic conductor: \(1.40137\)
Root analytic conductor: \(1.18379\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2808} (2755, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2808,\ (\ :0),\ 0.707 + 0.707i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.484292187\)
\(L(\frac12)\) \(\approx\) \(2.484292187\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.965 + 0.258i)T \)
3 \( 1 \)
13 \( 1 - iT \)
good5 \( 1 - 0.517T + T^{2} \)
7 \( 1 + T^{2} \)
11 \( 1 + 1.41iT - T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 - 1.41iT - T^{2} \)
43 \( 1 + T + T^{2} \)
47 \( 1 - 1.93T + T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + 0.517iT - T^{2} \)
61 \( 1 - iT - T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + 0.517T + T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 - 2iT - T^{2} \)
83 \( 1 + 1.93iT - T^{2} \)
89 \( 1 + 0.517iT - T^{2} \)
97 \( 1 - T^{2} \)
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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

−8.967045474670840692881657949905, −8.091983256007316529518430869520, −7.13878511558481328604910386302, −6.28962217117895485586077742455, −5.87588341962075707133259459309, −4.99552098727730720601281669841, −4.10495012970375627069854658311, −3.29317436772464335521652662864, −2.36898749830258388135025429498, −1.32302424600201149528249182834, 1.74587126767127505128596675033, 2.54044028150459939190878835658, 3.58688824961936792059569258256, 4.45811443661537610468070615506, 5.25864017615014487834302717828, 5.84990749280098790861767953013, 6.74248279079678015072820137058, 7.42054882081458120179268231687, 8.044024529312454462370458196507, 9.058485176878098029211367290214

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