Properties

Label 2-1008-21.5-c3-0-3
Degree $2$
Conductor $1008$
Sign $0.00687 - 0.999i$
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  + (7.54 − 13.0i)5-s + (−16.2 − 8.80i)7-s + (−8.56 + 4.94i)11-s + 67.8i·13-s + (−35.0 − 60.7i)17-s + (53.2 + 30.7i)19-s + (−113. − 65.7i)23-s + (−51.3 − 88.8i)25-s + 158. i·29-s + (66.2 − 38.2i)31-s + (−237. + 146. i)35-s + (−174. + 301. i)37-s − 138.·41-s − 539.·43-s + (111. − 193. i)47-s + ⋯
L(s)  = 1  + (0.674 − 1.16i)5-s + (−0.879 − 0.475i)7-s + (−0.234 + 0.135i)11-s + 1.44i·13-s + (−0.500 − 0.866i)17-s + (0.642 + 0.371i)19-s + (−1.03 − 0.596i)23-s + (−0.410 − 0.711i)25-s + 1.01i·29-s + (0.383 − 0.221i)31-s + (−1.14 + 0.707i)35-s + (−0.774 + 1.34i)37-s − 0.529·41-s − 1.91·43-s + (0.347 − 0.601i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.00687 - 0.999i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.00687 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $0.00687 - 0.999i$
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: \((2,\ 1008,\ (\ :3/2),\ 0.00687 - 0.999i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.8023540000\)
\(L(\frac12)\) \(\approx\) \(0.8023540000\)
\(L(\frac{5}{2})\) 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 \( 1 + (16.2 + 8.80i)T \)
good5 \( 1 + (-7.54 + 13.0i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (8.56 - 4.94i)T + (665.5 - 1.15e3i)T^{2} \)
13 \( 1 - 67.8iT - 2.19e3T^{2} \)
17 \( 1 + (35.0 + 60.7i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-53.2 - 30.7i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (113. + 65.7i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 - 158. iT - 2.43e4T^{2} \)
31 \( 1 + (-66.2 + 38.2i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (174. - 301. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + 138.T + 6.89e4T^{2} \)
43 \( 1 + 539.T + 7.95e4T^{2} \)
47 \( 1 + (-111. + 193. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (-459. + 265. i)T + (7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-271. - 470. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (116. + 67.0i)T + (1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-160. - 277. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 416. iT - 3.57e5T^{2} \)
73 \( 1 + (-472. + 272. i)T + (1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (161. - 279. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 - 885.T + 5.71e5T^{2} \)
89 \( 1 + (812. - 1.40e3i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + 739. iT - 9.12e5T^{2} \)
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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

−9.808280329919996473742396511629, −9.009062090271730999320084419866, −8.382385956039919155583350344477, −7.04567049214586337566318331136, −6.52057618823607088743683533917, −5.35416336257269281896691081480, −4.64001289305476631691448006383, −3.61118741938061576141615408759, −2.21632304431702596510494852634, −1.13108315112134765924231403713, 0.20329511529711058526391770846, 2.05922466842020152335237969274, 2.94417010839944640435688490311, 3.68772700283238590542661202675, 5.35217685232125175023123169575, 6.00281038154797560933616023978, 6.67507314319607763042093813789, 7.65104468678088146509038263487, 8.545036696638233528076531317182, 9.634133798215477003905924248323

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