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Properties

Label 2-1008-63.20-c1-0-17
Degree $2$
Conductor $1008$
Sign $-0.0248 - 0.999i$
Analytic cond. $8.04892$
Root an. cond. $2.83706$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.62 + 0.608i)3-s + (1.94 + 3.36i)5-s + (−0.343 + 2.62i)7-s + (2.26 + 1.97i)9-s + (−3.41 − 1.97i)11-s + (2.46 − 1.42i)13-s + (1.10 + 6.64i)15-s + 0.742·17-s − 1.78i·19-s + (−2.15 + 4.04i)21-s + (5.41 − 3.12i)23-s + (−5.07 + 8.78i)25-s + (2.46 + 4.57i)27-s + (−2.50 − 1.44i)29-s + (−3.04 + 1.75i)31-s + ⋯
L(s)  = 1  + (0.936 + 0.351i)3-s + (0.870 + 1.50i)5-s + (−0.130 + 0.991i)7-s + (0.753 + 0.657i)9-s + (−1.03 − 0.594i)11-s + (0.684 − 0.395i)13-s + (0.285 + 1.71i)15-s + 0.179·17-s − 0.409i·19-s + (−0.469 + 0.882i)21-s + (1.12 − 0.651i)23-s + (−1.01 + 1.75i)25-s + (0.474 + 0.880i)27-s + (−0.464 − 0.268i)29-s + (−0.546 + 0.315i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0248 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0248 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $-0.0248 - 0.999i$
Analytic conductor: \(8.04892\)
Root analytic conductor: \(2.83706\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (209, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1/2),\ -0.0248 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.484865788\)
\(L(\frac12)\) \(\approx\) \(2.484865788\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.62 - 0.608i)T \)
7 \( 1 + (0.343 - 2.62i)T \)
good5 \( 1 + (-1.94 - 3.36i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (3.41 + 1.97i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.46 + 1.42i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 - 0.742T + 17T^{2} \)
19 \( 1 + 1.78iT - 19T^{2} \)
23 \( 1 + (-5.41 + 3.12i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.50 + 1.44i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (3.04 - 1.75i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 3.00T + 37T^{2} \)
41 \( 1 + (5.24 + 9.08i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.471 - 0.816i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (1.09 - 1.89i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + (0.0105 + 0.0183i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.13 + 1.23i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-6.72 - 11.6i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 1.94iT - 71T^{2} \)
73 \( 1 + 4.85iT - 73T^{2} \)
79 \( 1 + (-1.81 + 3.14i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-4.02 + 6.98i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 9.26T + 89T^{2} \)
97 \( 1 + (-16.2 - 9.40i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.29294509671262517777174092562, −9.290293314815599997178314564798, −8.676455905264129719344042310339, −7.72563471574276810756989930171, −6.82259729277273430906069144460, −5.89418034362729417477987418997, −5.12542159108991889022631177775, −3.43527948659895330414929999455, −2.85844090765573479697046325491, −2.10832630161096757624000134277, 1.08887358077215962000637844667, 1.95349944303321116975672677681, 3.40292624997789076428631173255, 4.47531158969906080006766296579, 5.27675639844105808443705856812, 6.43126891244434410425594801905, 7.45700944810177033677329584966, 8.130605285582864794018463521779, 8.967313091396715544675043037762, 9.634945282025447913114751879264

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