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Properties

Label 2-1008-63.16-c1-0-5
Degree $2$
Conductor $1008$
Sign $0.176 - 0.984i$
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  + (−0.796 − 1.53i)3-s + 0.593·5-s + (0.0665 + 2.64i)7-s + (−1.73 + 2.45i)9-s + 0.593·11-s + (−1.25 − 2.17i)13-s + (−0.472 − 0.912i)15-s + (1.46 + 2.52i)17-s + (−2.69 + 4.66i)19-s + (4.01 − 2.20i)21-s − 4.46·23-s − 4.64·25-s + (5.14 + 0.708i)27-s + (−3.09 + 5.36i)29-s + (−3.93 + 6.81i)31-s + ⋯
L(s)  = 1  + (−0.460 − 0.887i)3-s + 0.265·5-s + (0.0251 + 0.999i)7-s + (−0.576 + 0.816i)9-s + 0.178·11-s + (−0.348 − 0.603i)13-s + (−0.122 − 0.235i)15-s + (0.354 + 0.613i)17-s + (−0.617 + 1.06i)19-s + (0.876 − 0.482i)21-s − 0.930·23-s − 0.929·25-s + (0.990 + 0.136i)27-s + (−0.575 + 0.996i)29-s + (−0.706 + 1.22i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.176 - 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.176 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $0.176 - 0.984i$
Analytic conductor: \(8.04892\)
Root analytic conductor: \(2.83706\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (961, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1/2),\ 0.176 - 0.984i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8598129210\)
\(L(\frac12)\) \(\approx\) \(0.8598129210\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.796 + 1.53i)T \)
7 \( 1 + (-0.0665 - 2.64i)T \)
good5 \( 1 - 0.593T + 5T^{2} \)
11 \( 1 - 0.593T + 11T^{2} \)
13 \( 1 + (1.25 + 2.17i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-1.46 - 2.52i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.69 - 4.66i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 4.46T + 23T^{2} \)
29 \( 1 + (3.09 - 5.36i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (3.93 - 6.81i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.5 + 0.866i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (0.136 + 0.236i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-5.58 + 9.66i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-6.08 - 10.5i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-4.02 - 6.97i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-4.32 + 7.48i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.32 - 5.75i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.956 - 1.65i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 14.4T + 71T^{2} \)
73 \( 1 + (-3.95 - 6.85i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (4.62 + 8.00i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (3.85 - 6.66i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (6.21 - 10.7i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-5.86 + 10.1i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.29612284310239568590028032009, −9.222522182613003464887846053694, −8.345954618909231653735611441732, −7.70141334975785682943786952041, −6.67992112372814921820878483390, −5.66897560981512336184785308792, −5.50332434734233590119635525496, −3.86311724165434886170667205590, −2.47782970599450940161830744477, −1.57466700014016380277034046154, 0.40613407761277491417661678431, 2.30040013057199497940038900502, 3.82998152510146496292193781024, 4.33031009736017941911610563167, 5.39100906960366712373194909806, 6.29953899247841360948737374285, 7.16886746610291347983425318985, 8.130430879906105467726882898732, 9.384647673467148961973404799231, 9.690371888485995994564725802550

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