Properties

Label 2-1008-21.5-c3-0-30
Degree $2$
Conductor $1008$
Sign $0.966 + 0.255i$
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  + (4.27 − 7.39i)5-s + (12.5 − 13.5i)7-s + (27.4 − 15.8i)11-s + 9.92i·13-s + (63.7 + 110. i)17-s + (100. + 58.2i)19-s + (−55.8 − 32.2i)23-s + (26.0 + 45.0i)25-s + 113. i·29-s + (−6.33 + 3.65i)31-s + (−46.8 − 151. i)35-s + (−184. + 319. i)37-s + 211.·41-s + 432.·43-s + (200. − 346. i)47-s + ⋯
L(s)  = 1  + (0.382 − 0.661i)5-s + (0.679 − 0.733i)7-s + (0.752 − 0.434i)11-s + 0.211i·13-s + (0.909 + 1.57i)17-s + (1.21 + 0.703i)19-s + (−0.506 − 0.292i)23-s + (0.208 + 0.360i)25-s + 0.723i·29-s + (−0.0367 + 0.0211i)31-s + (−0.226 − 0.729i)35-s + (−0.820 + 1.42i)37-s + 0.807·41-s + 1.53·43-s + (0.620 − 1.07i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.966 + 0.255i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.966 + 0.255i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $0.966 + 0.255i$
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: \((2,\ 1008,\ (\ :3/2),\ 0.966 + 0.255i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.871029223\)
\(L(\frac12)\) \(\approx\) \(2.871029223\)
\(L(\frac{5}{2})\) 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 \( 1 + (-12.5 + 13.5i)T \)
good5 \( 1 + (-4.27 + 7.39i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (-27.4 + 15.8i)T + (665.5 - 1.15e3i)T^{2} \)
13 \( 1 - 9.92iT - 2.19e3T^{2} \)
17 \( 1 + (-63.7 - 110. i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-100. - 58.2i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (55.8 + 32.2i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 - 113. iT - 2.43e4T^{2} \)
31 \( 1 + (6.33 - 3.65i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (184. - 319. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 - 211.T + 6.89e4T^{2} \)
43 \( 1 - 432.T + 7.95e4T^{2} \)
47 \( 1 + (-200. + 346. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (121. - 70.3i)T + (7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (259. + 449. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-23.5 - 13.6i)T + (1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (68.3 + 118. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 604. iT - 3.57e5T^{2} \)
73 \( 1 + (41.9 - 24.2i)T + (1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (415. - 719. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 - 37.2T + 5.71e5T^{2} \)
89 \( 1 + (-235. + 407. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 - 522. iT - 9.12e5T^{2} \)
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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

−9.574233774532214603523483793367, −8.636256045410569172060968725096, −8.009127523605078750564019980821, −7.08961259649630672691720813584, −5.98983636449378729423281903685, −5.27994196137425010578436929009, −4.18748817668358663481894550853, −3.41394132701892359333870033331, −1.61279334616616587279668463224, −1.07007521477820572056678683223, 0.916477429235834045490268798112, 2.26580945351676828448777110610, 3.05915783556812490525583106142, 4.41051862869942558408206228743, 5.39338885486497949798906528549, 6.09146559177775085628745751410, 7.29956328958627757006893881734, 7.68020852689100118157959061778, 9.127115796412409592648094021272, 9.401465146334398737472061291789

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