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Properties

Label 2-1008-63.16-c1-0-44
Degree $2$
Conductor $1008$
Sign $0.340 + 0.940i$
Analytic cond. $8.04892$
Root an. cond. $2.83706$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.40 − 1.01i)3-s + 3.84·5-s + (−0.676 − 2.55i)7-s + (0.953 − 2.84i)9-s − 1.80·11-s + (−0.692 − 1.19i)13-s + (5.40 − 3.88i)15-s + (−0.833 − 1.44i)17-s + (0.0802 − 0.138i)19-s + (−3.53 − 2.91i)21-s − 3.20·23-s + 9.75·25-s + (−1.53 − 4.96i)27-s + (−3.78 + 6.54i)29-s + (1.61 − 2.78i)31-s + ⋯
L(s)  = 1  + (0.811 − 0.584i)3-s + 1.71·5-s + (−0.255 − 0.966i)7-s + (0.317 − 0.948i)9-s − 0.544·11-s + (−0.192 − 0.332i)13-s + (1.39 − 1.00i)15-s + (−0.202 − 0.350i)17-s + (0.0184 − 0.0318i)19-s + (−0.772 − 0.635i)21-s − 0.667·23-s + 1.95·25-s + (−0.295 − 0.955i)27-s + (−0.701 + 1.21i)29-s + (0.289 − 0.500i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.340 + 0.940i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.340 + 0.940i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $0.340 + 0.940i$
Analytic conductor: \(8.04892\)
Root analytic conductor: \(2.83706\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (961, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1/2),\ 0.340 + 0.940i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.621654271\)
\(L(\frac12)\) \(\approx\) \(2.621654271\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.40 + 1.01i)T \)
7 \( 1 + (0.676 + 2.55i)T \)
good5 \( 1 - 3.84T + 5T^{2} \)
11 \( 1 + 1.80T + 11T^{2} \)
13 \( 1 + (0.692 + 1.19i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (0.833 + 1.44i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.0802 + 0.138i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 3.20T + 23T^{2} \)
29 \( 1 + (3.78 - 6.54i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-1.61 + 2.78i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.58 + 2.74i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-6.00 - 10.3i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.45 - 5.98i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-5.71 - 9.90i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.37 - 2.38i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-7.53 + 13.0i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.60 - 7.96i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.16 - 10.6i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 6.93T + 71T^{2} \)
73 \( 1 + (6.22 + 10.7i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-8.03 - 13.9i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-1.45 + 2.51i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-5.04 + 8.73i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-4.18 + 7.25i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.715556213633484015766060213090, −9.182993340716164016394876464676, −8.066023046979761151269490077316, −7.29426678075918671107304852128, −6.45080237537162775576923040729, −5.70316793285351694296332429506, −4.49391807358707043591259028696, −3.13482915826745854017416757707, −2.28154302857942164241841715685, −1.14480672403146749782837459905, 2.12235291012682873488472219470, 2.37796124152205917566384159567, 3.77215750139858273844214781495, 5.11132693554230505623289215776, 5.67811919196422396601799700260, 6.58709190740412887312952478534, 7.81260851199042986897053194714, 8.881521251069449894036667372001, 9.172996447937635510082576216525, 10.16774061415744238973707025648

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