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Properties

Label 2-1008-63.25-c1-0-5
Degree $2$
Conductor $1008$
Sign $-0.682 - 0.731i$
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  + (−1.57 − 0.711i)3-s + (−1.92 + 3.32i)5-s + (2.55 − 0.693i)7-s + (1.98 + 2.24i)9-s + (0.903 + 1.56i)11-s + (−0.692 − 1.19i)13-s + (5.40 − 3.88i)15-s + (−0.833 + 1.44i)17-s + (0.0802 + 0.138i)19-s + (−4.52 − 0.723i)21-s + (1.60 − 2.77i)23-s + (−4.87 − 8.44i)25-s + (−1.53 − 4.96i)27-s + (−3.78 + 6.54i)29-s − 3.22·31-s + ⋯
L(s)  = 1  + (−0.911 − 0.410i)3-s + (−0.858 + 1.48i)5-s + (0.965 − 0.261i)7-s + (0.662 + 0.749i)9-s + (0.272 + 0.471i)11-s + (−0.192 − 0.332i)13-s + (1.39 − 1.00i)15-s + (−0.202 + 0.350i)17-s + (0.0184 + 0.0318i)19-s + (−0.987 − 0.157i)21-s + (0.333 − 0.577i)23-s + (−0.975 − 1.68i)25-s + (−0.295 − 0.955i)27-s + (−0.701 + 1.21i)29-s − 0.578·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.682 - 0.731i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.682 - 0.731i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $-0.682 - 0.731i$
Analytic conductor: \(8.04892\)
Root analytic conductor: \(2.83706\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (529, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1/2),\ -0.682 - 0.731i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6600748489\)
\(L(\frac12)\) \(\approx\) \(0.6600748489\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.57 + 0.711i)T \)
7 \( 1 + (-2.55 + 0.693i)T \)
good5 \( 1 + (1.92 - 3.32i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-0.903 - 1.56i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.692 + 1.19i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (0.833 - 1.44i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.0802 - 0.138i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.60 + 2.77i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.78 - 6.54i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 3.22T + 31T^{2} \)
37 \( 1 + (-1.58 - 2.74i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-6.00 - 10.3i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.45 - 5.98i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 11.4T + 47T^{2} \)
53 \( 1 + (-1.37 + 2.38i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 15.0T + 59T^{2} \)
61 \( 1 + 9.20T + 61T^{2} \)
67 \( 1 - 12.3T + 67T^{2} \)
71 \( 1 - 6.93T + 71T^{2} \)
73 \( 1 + (6.22 - 10.7i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + 16.0T + 79T^{2} \)
83 \( 1 + (-1.45 + 2.51i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-5.04 - 8.73i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-4.18 + 7.25i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.64101164371618114434042147679, −9.732917413273546358883627648832, −8.205762428088558902786928714604, −7.61284167412045609113530976351, −6.92366766686683270063134591351, −6.26224591877692664050862793025, −5.00919955619186773523091058319, −4.20117622142616889356729837912, −2.96472387926134442880933231725, −1.57434722938102365934246090797, 0.35878440544325466741837972877, 1.63130717756839249759955137933, 3.75767620349619151509048045830, 4.50457368545763675346040027840, 5.17932233229653690768533701060, 5.90683713254523239578634254681, 7.28342150010950161139279098053, 8.011421322315318455186941349126, 9.022081281310994720590864243452, 9.370391241653990528913903758899

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