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Properties

Label 2-1008-21.5-c3-0-38
Degree $2$
Conductor $1008$
Sign $-0.839 + 0.542i$
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  + (−5.00 + 8.66i)5-s + (−1.56 − 18.4i)7-s + (−8.94 + 5.16i)11-s + 52.4i·13-s + (0.584 + 1.01i)17-s + (86.7 + 50.0i)19-s + (−90.1 − 52.0i)23-s + (12.4 + 21.6i)25-s − 187. i·29-s + (107. − 62.0i)31-s + (167. + 78.6i)35-s + (16.0 − 27.7i)37-s − 415.·41-s + 193.·43-s + (−196. + 340. i)47-s + ⋯
L(s)  = 1  + (−0.447 + 0.774i)5-s + (−0.0847 − 0.996i)7-s + (−0.245 + 0.141i)11-s + 1.11i·13-s + (0.00833 + 0.0144i)17-s + (1.04 + 0.604i)19-s + (−0.817 − 0.471i)23-s + (0.0999 + 0.173i)25-s − 1.20i·29-s + (0.622 − 0.359i)31-s + (0.809 + 0.379i)35-s + (0.0712 − 0.123i)37-s − 1.58·41-s + 0.685·43-s + (−0.609 + 1.05i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.839 + 0.542i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.839 + 0.542i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $-0.839 + 0.542i$
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: \((2,\ 1008,\ (\ :3/2),\ -0.839 + 0.542i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.1934262061\)
\(L(\frac12)\) \(\approx\) \(0.1934262061\)
\(L(\frac{5}{2})\) 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 \( 1 + (1.56 + 18.4i)T \)
good5 \( 1 + (5.00 - 8.66i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (8.94 - 5.16i)T + (665.5 - 1.15e3i)T^{2} \)
13 \( 1 - 52.4iT - 2.19e3T^{2} \)
17 \( 1 + (-0.584 - 1.01i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-86.7 - 50.0i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (90.1 + 52.0i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + 187. iT - 2.43e4T^{2} \)
31 \( 1 + (-107. + 62.0i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (-16.0 + 27.7i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + 415.T + 6.89e4T^{2} \)
43 \( 1 - 193.T + 7.95e4T^{2} \)
47 \( 1 + (196. - 340. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (74.7 - 43.1i)T + (7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-102. - 177. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (183. + 105. i)T + (1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (364. + 631. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 315. iT - 3.57e5T^{2} \)
73 \( 1 + (899. - 519. i)T + (1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-607. + 1.05e3i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 - 333.T + 5.71e5T^{2} \)
89 \( 1 + (168. - 291. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + 893. iT - 9.12e5T^{2} \)
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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

−9.471252841920720393246617567662, −8.180886567645902754310657495610, −7.51949039690515554162963492163, −6.81989028581439339602681281913, −5.99667945811829580536695900503, −4.61273926052460070649258015390, −3.88901783821970017232344910754, −2.91323999510549643872748679743, −1.54788767070263415331846040634, −0.05110218471436601780675954401, 1.21201840868466587617809298773, 2.67204338666186510338277477657, 3.56696223556055543474541507895, 5.00413586869551947588785722558, 5.35100202391625361769154724580, 6.47038356842575196990603037994, 7.62221852034596726335325159962, 8.347781382866454649860587224109, 8.941612645747229539395971068367, 9.855595007171952834849766583881

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