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Properties

Label 2-1008-21.5-c3-0-4
Degree $2$
Conductor $1008$
Sign $-0.216 - 0.976i$
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  + (7.89 − 13.6i)5-s + (−10.6 + 15.1i)7-s + (−28.2 + 16.3i)11-s − 54.3i·13-s + (12.3 + 21.3i)17-s + (−16.2 − 9.41i)19-s + (46.7 + 26.9i)23-s + (−62.0 − 107. i)25-s + 157. i·29-s + (−41.4 + 23.9i)31-s + (123. + 264. i)35-s + (−48.1 + 83.4i)37-s − 263.·41-s − 258.·43-s + (62.5 − 108. i)47-s + ⋯
L(s)  = 1  + (0.705 − 1.22i)5-s + (−0.573 + 0.819i)7-s + (−0.774 + 0.446i)11-s − 1.15i·13-s + (0.175 + 0.304i)17-s + (−0.196 − 0.113i)19-s + (0.423 + 0.244i)23-s + (−0.496 − 0.859i)25-s + 1.00i·29-s + (−0.240 + 0.138i)31-s + (0.596 + 1.27i)35-s + (−0.214 + 0.370i)37-s − 1.00·41-s − 0.918·43-s + (0.194 − 0.336i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.216 - 0.976i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.216 - 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $-0.216 - 0.976i$
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: \((2,\ 1008,\ (\ :3/2),\ -0.216 - 0.976i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.8703709441\)
\(L(\frac12)\) \(\approx\) \(0.8703709441\)
\(L(\frac{5}{2})\) 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 \( 1 + (10.6 - 15.1i)T \)
good5 \( 1 + (-7.89 + 13.6i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (28.2 - 16.3i)T + (665.5 - 1.15e3i)T^{2} \)
13 \( 1 + 54.3iT - 2.19e3T^{2} \)
17 \( 1 + (-12.3 - 21.3i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (16.2 + 9.41i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-46.7 - 26.9i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 - 157. iT - 2.43e4T^{2} \)
31 \( 1 + (41.4 - 23.9i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (48.1 - 83.4i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + 263.T + 6.89e4T^{2} \)
43 \( 1 + 258.T + 7.95e4T^{2} \)
47 \( 1 + (-62.5 + 108. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (471. - 272. i)T + (7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-189. - 328. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-587. - 339. i)T + (1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-346. - 600. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 238. iT - 3.57e5T^{2} \)
73 \( 1 + (-631. + 364. i)T + (1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (439. - 761. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + 1.22e3T + 5.71e5T^{2} \)
89 \( 1 + (75.9 - 131. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + 1.59e3iT - 9.12e5T^{2} \)
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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

−9.897133660554873373491054685290, −8.836517529481911642974523518410, −8.483162224594168656883444763260, −7.36336618978785534741505128390, −6.21238279151051474514264929963, −5.30372632642563489975829738435, −5.01430870664676306657483307213, −3.42043366072878930044513393846, −2.36504702150413937869639032069, −1.19029781780961658324574463434, 0.21520845747460893829513664757, 1.91271519633072845271532466474, 2.92437248722824912034677592786, 3.79433021639426767098541734590, 5.02762837321957858920498174730, 6.20944352520331644351516107906, 6.71509019903131108154854883458, 7.45755284668314086144371860165, 8.514082879224142632201442502480, 9.748352699758657829923907903125

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