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Properties

Label 2-1008-21.5-c3-0-45
Degree $2$
Conductor $1008$
Sign $-0.777 + 0.629i$
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  + (8.48 − 14.6i)5-s + (−3.52 − 18.1i)7-s + (60.3 − 34.8i)11-s − 38.2i·13-s + (−52.9 − 91.7i)17-s + (−50.3 − 29.0i)19-s + (107. + 62.3i)23-s + (−81.4 − 141. i)25-s − 66.5i·29-s + (136. − 78.8i)31-s + (−297. − 102. i)35-s + (−107. + 185. i)37-s + 448.·41-s + 320.·43-s + (−87.8 + 152. i)47-s + ⋯
L(s)  = 1  + (0.758 − 1.31i)5-s + (−0.190 − 0.981i)7-s + (1.65 − 0.955i)11-s − 0.815i·13-s + (−0.755 − 1.30i)17-s + (−0.608 − 0.351i)19-s + (0.978 + 0.564i)23-s + (−0.651 − 1.12i)25-s − 0.426i·29-s + (0.791 − 0.456i)31-s + (−1.43 − 0.494i)35-s + (−0.476 + 0.824i)37-s + 1.70·41-s + 1.13·43-s + (−0.272 + 0.472i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.777 + 0.629i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.777 + 0.629i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $-0.777 + 0.629i$
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: \((2,\ 1008,\ (\ :3/2),\ -0.777 + 0.629i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.557347531\)
\(L(\frac12)\) \(\approx\) \(2.557347531\)
\(L(\frac{5}{2})\) 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 \( 1 + (3.52 + 18.1i)T \)
good5 \( 1 + (-8.48 + 14.6i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (-60.3 + 34.8i)T + (665.5 - 1.15e3i)T^{2} \)
13 \( 1 + 38.2iT - 2.19e3T^{2} \)
17 \( 1 + (52.9 + 91.7i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (50.3 + 29.0i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-107. - 62.3i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + 66.5iT - 2.43e4T^{2} \)
31 \( 1 + (-136. + 78.8i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (107. - 185. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 - 448.T + 6.89e4T^{2} \)
43 \( 1 - 320.T + 7.95e4T^{2} \)
47 \( 1 + (87.8 - 152. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (585. - 338. i)T + (7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-343. - 594. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (91.1 + 52.6i)T + (1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-426. - 738. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 21.6iT - 3.57e5T^{2} \)
73 \( 1 + (-296. + 171. i)T + (1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-156. + 271. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + 627.T + 5.71e5T^{2} \)
89 \( 1 + (207. - 359. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 - 223. iT - 9.12e5T^{2} \)
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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

−9.202768834554659291178460967568, −8.712897819627305410576306370723, −7.60433801101236678331500102594, −6.59041136665735823245298234632, −5.85577700006024749970994915185, −4.78171783960093016648558640382, −4.08857199230206916841347772957, −2.80381904950209374531639115983, −1.15784914573934412277057665107, −0.71315225749320410884810495208, 1.73762430299769126250790958181, 2.33674387799199765975823031490, 3.59724420855328159364311791966, 4.57805999962281582996593040548, 5.99404507289228867544390540568, 6.55045122566179569494512291488, 6.96215752303161414969980806019, 8.451145128567737323311614989067, 9.239877623561081867448439894161, 9.752923238306097579872380208776

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