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Properties

Label 2-1008-63.16-c1-0-38
Degree $2$
Conductor $1008$
Sign $-0.568 + 0.822i$
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  + (0.377 − 1.69i)3-s + 1.42·5-s + (−2.21 − 1.44i)7-s + (−2.71 − 1.27i)9-s + 4.93·11-s + (−1.37 − 2.38i)13-s + (0.537 − 2.40i)15-s + (0.559 + 0.969i)17-s + (2.00 − 3.47i)19-s + (−3.28 + 3.19i)21-s − 5.43·23-s − 2.96·25-s + (−3.18 + 4.10i)27-s + (3.40 − 5.89i)29-s + (1.25 − 2.17i)31-s + ⋯
L(s)  = 1  + (0.217 − 0.975i)3-s + 0.637·5-s + (−0.837 − 0.546i)7-s + (−0.905 − 0.425i)9-s + 1.48·11-s + (−0.381 − 0.661i)13-s + (0.138 − 0.621i)15-s + (0.135 + 0.235i)17-s + (0.460 − 0.797i)19-s + (−0.716 + 0.698i)21-s − 1.13·23-s − 0.593·25-s + (−0.612 + 0.790i)27-s + (0.632 − 1.09i)29-s + (0.225 − 0.389i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.568 + 0.822i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.568 + 0.822i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $-0.568 + 0.822i$
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: \((2,\ 1008,\ (\ :1/2),\ -0.568 + 0.822i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.568495192\)
\(L(\frac12)\) \(\approx\) \(1.568495192\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.377 + 1.69i)T \)
7 \( 1 + (2.21 + 1.44i)T \)
good5 \( 1 - 1.42T + 5T^{2} \)
11 \( 1 - 4.93T + 11T^{2} \)
13 \( 1 + (1.37 + 2.38i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-0.559 - 0.969i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.00 + 3.47i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 5.43T + 23T^{2} \)
29 \( 1 + (-3.40 + 5.89i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-1.25 + 2.17i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.709 + 1.22i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-0.124 - 0.215i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.498 + 0.863i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (4.73 + 8.20i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.410 + 0.710i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (3.29 - 5.70i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.0376 + 0.0651i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.29 - 10.9i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 0.0804T + 71T^{2} \)
73 \( 1 + (-5.34 - 9.25i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (0.922 + 1.59i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-7.23 + 12.5i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-6.76 + 11.7i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-2.70 + 4.67i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.675127790143106607895903368388, −8.897779868623910203720858458869, −7.893287918873688227400823191427, −7.07024500137347185950477299616, −6.31258445268732486360032327284, −5.76046822213339747540224721309, −4.18858431588774062764997349701, −3.15534550613492811104905498410, −2.01109948051545980630094583635, −0.69200552382837875823250917582, 1.83760714900696141089222377626, 3.12451248610896405797040172432, 3.95283365257708362894399029385, 5.00758607157225014860124876103, 6.05178419280100647546357381074, 6.55311089884045977050801951065, 7.937916274477313846597649011898, 8.977069615535660139784506804785, 9.528871179706777877645200810607, 9.882972931619347433220748918429

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