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Properties

Label 2-1008-63.58-c1-0-17
Degree $2$
Conductor $1008$
Sign $0.0464 - 0.998i$
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  + (0.455 + 1.67i)3-s + (0.240 + 0.416i)5-s + (1.92 − 1.81i)7-s + (−2.58 + 1.52i)9-s + (1.69 − 2.92i)11-s + (−2.86 + 4.95i)13-s + (−0.587 + 0.592i)15-s + (2.75 + 4.77i)17-s + (−2.18 + 3.77i)19-s + (3.90 + 2.39i)21-s + (1.81 + 3.14i)23-s + (2.38 − 4.12i)25-s + (−3.71 − 3.62i)27-s + (1.53 + 2.65i)29-s + 9.34·31-s + ⋯
L(s)  = 1  + (0.262 + 0.964i)3-s + (0.107 + 0.186i)5-s + (0.728 − 0.684i)7-s + (−0.861 + 0.507i)9-s + (0.509 − 0.882i)11-s + (−0.793 + 1.37i)13-s + (−0.151 + 0.152i)15-s + (0.668 + 1.15i)17-s + (−0.500 + 0.866i)19-s + (0.852 + 0.522i)21-s + (0.378 + 0.654i)23-s + (0.476 − 0.825i)25-s + (−0.715 − 0.698i)27-s + (0.284 + 0.492i)29-s + 1.67·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0464 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0464 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $0.0464 - 0.998i$
Analytic conductor: \(8.04892\)
Root analytic conductor: \(2.83706\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (625, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1/2),\ 0.0464 - 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.813895602\)
\(L(\frac12)\) \(\approx\) \(1.813895602\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.455 - 1.67i)T \)
7 \( 1 + (-1.92 + 1.81i)T \)
good5 \( 1 + (-0.240 - 0.416i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.69 + 2.92i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (2.86 - 4.95i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-2.75 - 4.77i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.18 - 3.77i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.81 - 3.14i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.53 - 2.65i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 9.34T + 31T^{2} \)
37 \( 1 + (-1.48 + 2.57i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (6.29 - 10.9i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.90 + 3.30i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 3.76T + 47T^{2} \)
53 \( 1 + (-5.57 - 9.66i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + 8.42T + 59T^{2} \)
61 \( 1 + 7.28T + 61T^{2} \)
67 \( 1 + 2.57T + 67T^{2} \)
71 \( 1 - 3.94T + 71T^{2} \)
73 \( 1 + (0.862 + 1.49i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 - 5.59T + 79T^{2} \)
83 \( 1 + (-0.119 - 0.206i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-0.648 + 1.12i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (7.02 + 12.1i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.26511759366407329704742720457, −9.404300681586833325543114908514, −8.498864110904275351096409724050, −7.940718784592278903785366981487, −6.71348062441072788540461140148, −5.84359195342085277354651272868, −4.63826935310822625773656127044, −4.09803833214536758597549854441, −3.02981543629401483000974866683, −1.56134398541828787041858788398, 0.865625624746330389148849504103, 2.26235885615271799936390518290, 2.99632622701526656872941174634, 4.77043760239972407306643435096, 5.34193224984671962494042682832, 6.52981983573093534418968042251, 7.32026676370057469313743825303, 8.039706021394538490868375877987, 8.822179763031672494140598270309, 9.595398280280161780827852934760

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