Properties

Label 2-1008-21.17-c3-0-6
Degree $2$
Conductor $1008$
Sign $-0.997 - 0.0668i$
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  + (0.632 + 1.09i)5-s + (−13.4 + 12.7i)7-s + (36.0 + 20.7i)11-s + 85.7i·13-s + (−38.8 + 67.3i)17-s + (−42.1 + 24.3i)19-s + (78.7 − 45.4i)23-s + (61.6 − 106. i)25-s − 151. i·29-s + (−76.3 − 44.0i)31-s + (−22.4 − 6.60i)35-s + (−45.2 − 78.4i)37-s − 383.·41-s + 227.·43-s + (69.5 + 120. i)47-s + ⋯
L(s)  = 1  + (0.0566 + 0.0980i)5-s + (−0.723 + 0.690i)7-s + (0.987 + 0.570i)11-s + 1.82i·13-s + (−0.554 + 0.961i)17-s + (−0.509 + 0.293i)19-s + (0.714 − 0.412i)23-s + (0.493 − 0.854i)25-s − 0.968i·29-s + (−0.442 − 0.255i)31-s + (−0.108 − 0.0318i)35-s + (−0.201 − 0.348i)37-s − 1.46·41-s + 0.808·43-s + (0.215 + 0.373i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 - 0.0668i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.997 - 0.0668i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $-0.997 - 0.0668i$
Analytic conductor: \(59.4739\)
Root analytic conductor: \(7.71193\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :3/2),\ -0.997 - 0.0668i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.8889763708\)
\(L(\frac12)\) \(\approx\) \(0.8889763708\)
\(L(\frac{5}{2})\) 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 \( 1 + (13.4 - 12.7i)T \)
good5 \( 1 + (-0.632 - 1.09i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (-36.0 - 20.7i)T + (665.5 + 1.15e3i)T^{2} \)
13 \( 1 - 85.7iT - 2.19e3T^{2} \)
17 \( 1 + (38.8 - 67.3i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (42.1 - 24.3i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-78.7 + 45.4i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + 151. iT - 2.43e4T^{2} \)
31 \( 1 + (76.3 + 44.0i)T + (1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (45.2 + 78.4i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + 383.T + 6.89e4T^{2} \)
43 \( 1 - 227.T + 7.95e4T^{2} \)
47 \( 1 + (-69.5 - 120. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-289. - 167. i)T + (7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (440. - 762. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-11.3 + 6.57i)T + (1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (221. - 383. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 341. iT - 3.57e5T^{2} \)
73 \( 1 + (798. + 460. i)T + (1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (206. + 357. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 954.T + 5.71e5T^{2} \)
89 \( 1 + (-14.8 - 25.7i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 - 1.19e3iT - 9.12e5T^{2} \)
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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

−9.894788068819000773438489844853, −8.931392363439245394090144077956, −8.791545314461599187080254376014, −7.25790219130142472825773317264, −6.50170347561811876665474288497, −6.02993678770172618933274247245, −4.51233323569072903346989474082, −3.94639679367436815437723043304, −2.51494680967681846287709656601, −1.62089161398446707522444157410, 0.23049720313961390378609315516, 1.22052193089095717599548625978, 2.99292262770777617032276037297, 3.54601828344215048084023863044, 4.85093569973159004631958889095, 5.70857858817091301000630002101, 6.79326419294934410705220348633, 7.28799114668493951015052054663, 8.500342519474918789206780077290, 9.141512611327592694128206451434

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