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Properties

Label 2-1008-336.317-c0-0-1
Degree $2$
Conductor $1008$
Sign $0.631 - 0.775i$
Analytic cond. $0.503057$
Root an. cond. $0.709265$
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 + (−0.866 + 0.499i)4-s + (0.965 − 0.258i)5-s + (0.866 − 0.5i)7-s + (−0.707 − 0.707i)8-s + (0.499 + 0.866i)10-s + (0.258 − 0.965i)11-s + (0.707 + 0.707i)14-s + (0.500 − 0.866i)16-s + (−1.22 + 0.707i)17-s + (1.36 − 0.366i)19-s + (−0.707 + 0.707i)20-s + 22-s + (−0.5 + 0.866i)28-s + (−0.707 + 0.707i)29-s + ⋯
L(s)  = 1  + (0.258 + 0.965i)2-s + (−0.866 + 0.499i)4-s + (0.965 − 0.258i)5-s + (0.866 − 0.5i)7-s + (−0.707 − 0.707i)8-s + (0.499 + 0.866i)10-s + (0.258 − 0.965i)11-s + (0.707 + 0.707i)14-s + (0.500 − 0.866i)16-s + (−1.22 + 0.707i)17-s + (1.36 − 0.366i)19-s + (−0.707 + 0.707i)20-s + 22-s + (−0.5 + 0.866i)28-s + (−0.707 + 0.707i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.631 - 0.775i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.631 - 0.775i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $0.631 - 0.775i$
Analytic conductor: \(0.503057\)
Root analytic conductor: \(0.709265\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (989, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :0),\ 0.631 - 0.775i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.309842657\)
\(L(\frac12)\) \(\approx\) \(1.309842657\)
\(L(1)\) 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 \( 1 \)
7 \( 1 + (-0.866 + 0.5i)T \)
good5 \( 1 + (-0.965 + 0.258i)T + (0.866 - 0.5i)T^{2} \)
11 \( 1 + (-0.258 + 0.965i)T + (-0.866 - 0.5i)T^{2} \)
13 \( 1 + iT^{2} \)
17 \( 1 + (1.22 - 0.707i)T + (0.5 - 0.866i)T^{2} \)
19 \( 1 + (-1.36 + 0.366i)T + (0.866 - 0.5i)T^{2} \)
23 \( 1 + (-0.5 - 0.866i)T^{2} \)
29 \( 1 + (0.707 - 0.707i)T - iT^{2} \)
31 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
37 \( 1 + (-0.366 - 1.36i)T + (-0.866 + 0.5i)T^{2} \)
41 \( 1 + 1.41T + T^{2} \)
43 \( 1 + (1 - i)T - iT^{2} \)
47 \( 1 + (0.5 + 0.866i)T^{2} \)
53 \( 1 + (0.258 - 0.965i)T + (-0.866 - 0.5i)T^{2} \)
59 \( 1 + (0.258 - 0.965i)T + (-0.866 - 0.5i)T^{2} \)
61 \( 1 + (0.866 - 0.5i)T^{2} \)
67 \( 1 + (-0.866 - 0.5i)T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (0.5 - 0.866i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + (-0.707 + 0.707i)T - iT^{2} \)
89 \( 1 + (-0.707 + 1.22i)T + (-0.5 - 0.866i)T^{2} \)
97 \( 1 - T + T^{2} \)
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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

−10.06757420971277749082244189722, −9.225267754007056738958358226446, −8.568402741194561682574078428220, −7.76010052163105206549520578857, −6.81990195386769584829310507665, −5.97304909340462106930068501200, −5.24078137735542558934045352603, −4.39162977064372153874124114183, −3.23687082751312733694225799043, −1.49853715802956434103445936430, 1.76166584827434724853059176492, 2.31095785793344059921823399316, 3.67777513871499827172391819859, 4.91345079415074490531731718092, 5.37342564141962102016838215004, 6.49771137640706309712773981913, 7.60115896244411955328668260716, 8.774098138965080266766415032291, 9.436045719139351844541673374001, 10.01849073176975382608813054725

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