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Properties

Label 2-1008-28.27-c3-0-0
Degree $2$
Conductor $1008$
Sign $-0.803 + 0.594i$
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  + 17.7i·5-s + (−2.09 − 18.4i)7-s + 32.9i·11-s + 53.7i·13-s + 48.1i·17-s − 110.·19-s − 157. i·23-s − 189.·25-s + 72.0·29-s − 184.·31-s + (326. − 37.1i)35-s + 422.·37-s + 346. i·41-s − 198. i·43-s − 29.0·47-s + ⋯
L(s)  = 1  + 1.58i·5-s + (−0.113 − 0.993i)7-s + 0.902i·11-s + 1.14i·13-s + 0.686i·17-s − 1.33·19-s − 1.43i·23-s − 1.51·25-s + 0.461·29-s − 1.06·31-s + (1.57 − 0.179i)35-s + 1.87·37-s + 1.31i·41-s − 0.702i·43-s − 0.0902·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.803 + 0.594i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.803 + 0.594i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $-0.803 + 0.594i$
Analytic conductor: \(59.4739\)
Root analytic conductor: \(7.71193\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (559, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :3/2),\ -0.803 + 0.594i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.3649686718\)
\(L(\frac12)\) \(\approx\) \(0.3649686718\)
\(L(\frac{5}{2})\) 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 \( 1 + (2.09 + 18.4i)T \)
good5 \( 1 - 17.7iT - 125T^{2} \)
11 \( 1 - 32.9iT - 1.33e3T^{2} \)
13 \( 1 - 53.7iT - 2.19e3T^{2} \)
17 \( 1 - 48.1iT - 4.91e3T^{2} \)
19 \( 1 + 110.T + 6.85e3T^{2} \)
23 \( 1 + 157. iT - 1.21e4T^{2} \)
29 \( 1 - 72.0T + 2.43e4T^{2} \)
31 \( 1 + 184.T + 2.97e4T^{2} \)
37 \( 1 - 422.T + 5.06e4T^{2} \)
41 \( 1 - 346. iT - 6.89e4T^{2} \)
43 \( 1 + 198. iT - 7.95e4T^{2} \)
47 \( 1 + 29.0T + 1.03e5T^{2} \)
53 \( 1 + 682.T + 1.48e5T^{2} \)
59 \( 1 - 305.T + 2.05e5T^{2} \)
61 \( 1 + 172. iT - 2.26e5T^{2} \)
67 \( 1 + 109. iT - 3.00e5T^{2} \)
71 \( 1 - 741. iT - 3.57e5T^{2} \)
73 \( 1 + 752. iT - 3.89e5T^{2} \)
79 \( 1 - 449. iT - 4.93e5T^{2} \)
83 \( 1 - 262.T + 5.71e5T^{2} \)
89 \( 1 + 674. iT - 7.04e5T^{2} \)
97 \( 1 + 1.45e3iT - 9.12e5T^{2} \)
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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

−10.19581865586215353758496435666, −9.506890653327483902745917420696, −8.266214100273545972081770865592, −7.38907816698108275216296104184, −6.58690102271752729810917888165, −6.35653925187458161416969455842, −4.51150538430881439060594071193, −3.98433047287609823118828250664, −2.74420425906499074818310916376, −1.78914500997470519147314885813, 0.092778977568969205774594739992, 1.17885106174725199314245160975, 2.50774821514057327942144500266, 3.70031497555661596419507821099, 4.91147827777155804417478630440, 5.52673361764681373231010174068, 6.20927033554762539106849708563, 7.76019075122470055183559974708, 8.326527723475535400013043589522, 9.092745218279860458562115587424

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