Properties

Degree $2$
Conductor $1008$
Sign $-1$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.70 − 0.300i)3-s + (−1.26 − 2.19i)5-s + (0.5 − 0.866i)7-s + (2.81 + 1.02i)9-s + (0.233 − 0.405i)11-s + (−2.91 − 5.04i)13-s + (1.5 + 4.12i)15-s + 3.87·17-s + 2.18·19-s + (−1.11 + 1.32i)21-s + (−0.0530 − 0.0918i)23-s + (−0.705 + 1.22i)25-s + (−4.49 − 2.59i)27-s + (−4.39 + 7.60i)29-s + (−3.84 − 6.65i)31-s + ⋯
L(s)  = 1  + (−0.984 − 0.173i)3-s + (−0.566 − 0.980i)5-s + (0.188 − 0.327i)7-s + (0.939 + 0.342i)9-s + (0.0705 − 0.122i)11-s + (−0.807 − 1.39i)13-s + (0.387 + 1.06i)15-s + 0.940·17-s + 0.501·19-s + (−0.242 + 0.289i)21-s + (−0.0110 − 0.0191i)23-s + (−0.141 + 0.244i)25-s + (−0.866 − 0.499i)27-s + (−0.815 + 1.41i)29-s + (−0.689 − 1.19i)31-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{1008} (673, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4969853143\)
\(L(\frac12)\) \(\approx\) \(0.4969853143\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.70 + 0.300i)T \)
7 \( 1 + (-0.5 + 0.866i)T \)
good5 \( 1 + (1.26 + 2.19i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-0.233 + 0.405i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (2.91 + 5.04i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 - 3.87T + 17T^{2} \)
19 \( 1 - 2.18T + 19T^{2} \)
23 \( 1 + (0.0530 + 0.0918i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (4.39 - 7.60i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (3.84 + 6.65i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 7.68T + 37T^{2} \)
41 \( 1 + (-1.11 - 1.92i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.613 + 1.06i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (2.66 - 4.61i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 0.716T + 53T^{2} \)
59 \( 1 + (-0.368 - 0.637i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.479 - 0.829i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.81 + 8.34i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 13.2T + 71T^{2} \)
73 \( 1 + 10.2T + 73T^{2} \)
79 \( 1 + (6.31 - 10.9i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (1.36 - 2.36i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 8.11T + 89T^{2} \)
97 \( 1 + (-6.80 + 11.7i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.725034417280832606976108675359, −8.630592519807894043189613004347, −7.63194189944478149141349563422, −7.30289454772301239431520128419, −5.81658589064262948397060554976, −5.25951120974827262824113446218, −4.47119294362224353242174637234, −3.29895697477240429701127352211, −1.40612509920672158778229391881, −0.27427708269657519555789863141, 1.78617183941398144140853609023, 3.28058416340649369229655902088, 4.26717112675753584662117407788, 5.22970001412823381162898142837, 6.13373739265818901579652288431, 7.20032853034883521802606103902, 7.37387828310427735713459142139, 8.858139447915015586790874563810, 9.775609260244395990654377005278, 10.41230381473354823538142171521

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