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Properties

Label 2-1008-48.35-c1-0-34
Degree $2$
Conductor $1008$
Sign $0.937 - 0.347i$
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  + (1.37 + 0.347i)2-s + (1.75 + 0.952i)4-s + (−0.111 − 0.111i)5-s + 7-s + (2.07 + 1.91i)8-s + (−0.114 − 0.191i)10-s + (3.61 − 3.61i)11-s + (−1.94 − 1.94i)13-s + (1.37 + 0.347i)14-s + (2.18 + 3.35i)16-s + 4.79i·17-s + (3.03 − 3.03i)19-s + (−0.0897 − 0.302i)20-s + (6.20 − 3.69i)22-s + 6.58i·23-s + ⋯
L(s)  = 1  + (0.969 + 0.245i)2-s + (0.879 + 0.476i)4-s + (−0.0498 − 0.0498i)5-s + 0.377·7-s + (0.735 + 0.677i)8-s + (−0.0360 − 0.0605i)10-s + (1.08 − 1.08i)11-s + (−0.539 − 0.539i)13-s + (0.366 + 0.0928i)14-s + (0.546 + 0.837i)16-s + 1.16i·17-s + (0.695 − 0.695i)19-s + (−0.0200 − 0.0675i)20-s + (1.32 − 0.788i)22-s + 1.37i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.937 - 0.347i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.937 - 0.347i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $0.937 - 0.347i$
Analytic conductor: \(8.04892\)
Root analytic conductor: \(2.83706\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (323, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1/2),\ 0.937 - 0.347i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.241542273\)
\(L(\frac12)\) \(\approx\) \(3.241542273\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.37 - 0.347i)T \)
3 \( 1 \)
7 \( 1 - T \)
good5 \( 1 + (0.111 + 0.111i)T + 5iT^{2} \)
11 \( 1 + (-3.61 + 3.61i)T - 11iT^{2} \)
13 \( 1 + (1.94 + 1.94i)T + 13iT^{2} \)
17 \( 1 - 4.79iT - 17T^{2} \)
19 \( 1 + (-3.03 + 3.03i)T - 19iT^{2} \)
23 \( 1 - 6.58iT - 23T^{2} \)
29 \( 1 + (-1.53 + 1.53i)T - 29iT^{2} \)
31 \( 1 - 3.26iT - 31T^{2} \)
37 \( 1 + (-1.05 + 1.05i)T - 37iT^{2} \)
41 \( 1 + 1.26T + 41T^{2} \)
43 \( 1 + (-0.484 - 0.484i)T + 43iT^{2} \)
47 \( 1 + 11.2T + 47T^{2} \)
53 \( 1 + (-4.00 - 4.00i)T + 53iT^{2} \)
59 \( 1 + (7.61 - 7.61i)T - 59iT^{2} \)
61 \( 1 + (-5.44 - 5.44i)T + 61iT^{2} \)
67 \( 1 + (0.897 - 0.897i)T - 67iT^{2} \)
71 \( 1 - 2.83iT - 71T^{2} \)
73 \( 1 + 15.7iT - 73T^{2} \)
79 \( 1 + 15.4iT - 79T^{2} \)
83 \( 1 + (7.57 + 7.57i)T + 83iT^{2} \)
89 \( 1 - 13.1T + 89T^{2} \)
97 \( 1 + 10.4T + 97T^{2} \)
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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

−10.21253459170271093037905716886, −9.012559894220876219726968545168, −8.184756685403063109394023631639, −7.41278466759342619008203020164, −6.39326550860314895947563093863, −5.73215949074780299190583760278, −4.77328732952723708854015733287, −3.78942949687504357775809196919, −2.94039190550214937527690944843, −1.44519168101339132870353666056, 1.44791285681641722255307886889, 2.53800516235328119932742148681, 3.78155629384934498240161703692, 4.64618699035593809505245793976, 5.31518956503397593764839860770, 6.65465539180193336527595129732, 7.02847124278632703082117078351, 8.067529668863667604215525919610, 9.489364273118675161292556915110, 9.812507205296599539080500944229

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