Properties

Label 2-1008-21.5-c3-0-40
Degree $2$
Conductor $1008$
Sign $-0.548 + 0.836i$
Analytic cond. $59.4739$
Root an. cond. $7.71193$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.08 − 3.60i)5-s + (18.2 − 2.97i)7-s + (33.1 − 19.1i)11-s − 49.6i·13-s + (−7.65 − 13.2i)17-s + (−122. − 70.9i)19-s + (−136. − 78.9i)23-s + (53.8 + 93.2i)25-s − 204. i·29-s + (90.5 − 52.2i)31-s + (27.3 − 72.1i)35-s + (−194. + 336. i)37-s + 325.·41-s − 191.·43-s + (−249. + 432. i)47-s + ⋯
L(s)  = 1  + (0.186 − 0.322i)5-s + (0.986 − 0.160i)7-s + (0.909 − 0.524i)11-s − 1.05i·13-s + (−0.109 − 0.189i)17-s + (−1.48 − 0.856i)19-s + (−1.24 − 0.715i)23-s + (0.430 + 0.745i)25-s − 1.31i·29-s + (0.524 − 0.302i)31-s + (0.131 − 0.348i)35-s + (−0.862 + 1.49i)37-s + 1.23·41-s − 0.678·43-s + (−0.775 + 1.34i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.548 + 0.836i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.548 + 0.836i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $-0.548 + 0.836i$
Analytic conductor: \(59.4739\)
Root analytic conductor: \(7.71193\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (593, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :3/2),\ -0.548 + 0.836i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.805059973\)
\(L(\frac12)\) \(\approx\) \(1.805059973\)
\(L(\frac{5}{2})\) 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 \( 1 \)
7 \( 1 + (-18.2 + 2.97i)T \)
good5 \( 1 + (-2.08 + 3.60i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (-33.1 + 19.1i)T + (665.5 - 1.15e3i)T^{2} \)
13 \( 1 + 49.6iT - 2.19e3T^{2} \)
17 \( 1 + (7.65 + 13.2i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (122. + 70.9i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (136. + 78.9i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + 204. iT - 2.43e4T^{2} \)
31 \( 1 + (-90.5 + 52.2i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (194. - 336. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 - 325.T + 6.89e4T^{2} \)
43 \( 1 + 191.T + 7.95e4T^{2} \)
47 \( 1 + (249. - 432. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (-37.8 + 21.8i)T + (7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-86.5 - 149. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-208. - 120. i)T + (1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (440. + 763. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 1.01e3iT - 3.57e5T^{2} \)
73 \( 1 + (361. - 208. i)T + (1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (237. - 411. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + 652.T + 5.71e5T^{2} \)
89 \( 1 + (-298. + 517. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 - 1.77e3iT - 9.12e5T^{2} \)
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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

−9.191214743268108663768911833913, −8.378576848009440667831427976426, −7.891970124577835785855130331896, −6.62971387915417726839985174072, −5.91442772979557206098967994585, −4.80077841066117245298179660995, −4.14028566551149072385370659536, −2.77526351252379080179330911720, −1.57613939628257034894626680665, −0.43089977143013683550161206752, 1.56366115121976316850016375870, 2.16335438704687857609837456043, 3.85795304564655784111381832071, 4.44440954245558136558158125491, 5.62928769952321773080113632540, 6.55069745265369616735061305175, 7.24022616142347307192433419230, 8.398000657288273016673188308502, 8.867876446076707233718586923541, 9.957557072656829274558725731788

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