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Properties

Label 2-1008-21.5-c3-0-44
Degree $2$
Conductor $1008$
Sign $-0.999 - 0.00279i$
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  + (−1.35 + 2.35i)5-s + (8.74 − 16.3i)7-s + (−8.39 + 4.84i)11-s − 67.7i·13-s + (−50.1 − 86.7i)17-s + (59.6 + 34.4i)19-s + (−126. − 73.3i)23-s + (58.8 + 101. i)25-s + 284. i·29-s + (−197. + 113. i)31-s + (26.5 + 42.7i)35-s + (150. − 261. i)37-s + 232.·41-s − 173.·43-s + (−191. + 331. i)47-s + ⋯
L(s)  = 1  + (−0.121 + 0.210i)5-s + (0.471 − 0.881i)7-s + (−0.229 + 0.132i)11-s − 1.44i·13-s + (−0.714 − 1.23i)17-s + (0.720 + 0.416i)19-s + (−1.15 − 0.664i)23-s + (0.470 + 0.814i)25-s + 1.82i·29-s + (−1.14 + 0.659i)31-s + (0.128 + 0.206i)35-s + (0.669 − 1.16i)37-s + 0.885·41-s − 0.613·43-s + (−0.594 + 1.02i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.00279i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.00279i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $-0.999 - 0.00279i$
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: \((2,\ 1008,\ (\ :3/2),\ -0.999 - 0.00279i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.5006461787\)
\(L(\frac12)\) \(\approx\) \(0.5006461787\)
\(L(\frac{5}{2})\) 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 \( 1 + (-8.74 + 16.3i)T \)
good5 \( 1 + (1.35 - 2.35i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (8.39 - 4.84i)T + (665.5 - 1.15e3i)T^{2} \)
13 \( 1 + 67.7iT - 2.19e3T^{2} \)
17 \( 1 + (50.1 + 86.7i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-59.6 - 34.4i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (126. + 73.3i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 - 284. iT - 2.43e4T^{2} \)
31 \( 1 + (197. - 113. i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (-150. + 261. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 - 232.T + 6.89e4T^{2} \)
43 \( 1 + 173.T + 7.95e4T^{2} \)
47 \( 1 + (191. - 331. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (-22.2 + 12.8i)T + (7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (371. + 644. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (343. + 198. i)T + (1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-293. - 508. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 501. iT - 3.57e5T^{2} \)
73 \( 1 + (-616. + 356. i)T + (1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (466. - 807. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + 837.T + 5.71e5T^{2} \)
89 \( 1 + (442. - 766. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + 1.17e3iT - 9.12e5T^{2} \)
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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

−9.234799335021155249129974257559, −8.160018679459703331146972449479, −7.48781445028996261245527063676, −6.85735837599134645677479899565, −5.53712181897861057063299089000, −4.86662928217174892806346258324, −3.70282402920594846504298826286, −2.79390425097919594640164415986, −1.33230947263639794842770087623, −0.12250449769442346571440497078, 1.63807431518090193020289664610, 2.46163831404266657632915133115, 3.96202078612277890570192160487, 4.66891256297917177613894643386, 5.83185727388075711845925755886, 6.43314909137710560556647103971, 7.67915106006701598539929010508, 8.354629105401294620154756601549, 9.145455089922006554211909240809, 9.825777250219455767655439648803

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