Properties

Label 2-1008-63.25-c1-0-8
Degree $2$
Conductor $1008$
Sign $-0.512 - 0.858i$
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  + (−1.64 + 0.553i)3-s + (−0.263 + 0.455i)5-s + (0.333 + 2.62i)7-s + (2.38 − 1.81i)9-s + (2.30 + 3.99i)11-s + (0.244 + 0.423i)13-s + (0.179 − 0.893i)15-s + (2.75 − 4.77i)17-s + (−1.83 − 3.18i)19-s + (−1.99 − 4.12i)21-s + (−0.0269 + 0.0467i)23-s + (2.36 + 4.09i)25-s + (−2.91 + 4.30i)27-s + (−3.28 + 5.68i)29-s − 6.07·31-s + ⋯
L(s)  = 1  + (−0.947 + 0.319i)3-s + (−0.117 + 0.203i)5-s + (0.125 + 0.992i)7-s + (0.796 − 0.605i)9-s + (0.695 + 1.20i)11-s + (0.0678 + 0.117i)13-s + (0.0464 − 0.230i)15-s + (0.668 − 1.15i)17-s + (−0.421 − 0.730i)19-s + (−0.436 − 0.899i)21-s + (−0.00562 + 0.00974i)23-s + (0.472 + 0.818i)25-s + (−0.561 + 0.827i)27-s + (−0.609 + 1.05i)29-s − 1.09·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.512 - 0.858i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.512 - 0.858i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $-0.512 - 0.858i$
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: \((2,\ 1008,\ (\ :1/2),\ -0.512 - 0.858i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9604842818\)
\(L(\frac12)\) \(\approx\) \(0.9604842818\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.64 - 0.553i)T \)
7 \( 1 + (-0.333 - 2.62i)T \)
good5 \( 1 + (0.263 - 0.455i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-2.30 - 3.99i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.244 - 0.423i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-2.75 + 4.77i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.83 + 3.18i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.0269 - 0.0467i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.28 - 5.68i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 6.07T + 31T^{2} \)
37 \( 1 + (-0.223 - 0.387i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-2.52 - 4.36i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.84 - 4.93i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 9.19T + 47T^{2} \)
53 \( 1 + (4.37 - 7.57i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 6.63T + 59T^{2} \)
61 \( 1 + 0.465T + 61T^{2} \)
67 \( 1 + 5.19T + 67T^{2} \)
71 \( 1 + 1.76T + 71T^{2} \)
73 \( 1 + (5.23 - 9.07i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + 16.3T + 79T^{2} \)
83 \( 1 + (4.49 - 7.78i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (7.05 + 12.2i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-5.22 + 9.04i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.23032256840026152759294484489, −9.362255589248797127756009142292, −8.968955850563216596993718651847, −7.38203768306208305457792188992, −6.94154991204590920663831792363, −5.83678852157581037131649093646, −5.09013299347607631643254489439, −4.28465751653569743747022500909, −2.96492762516451866555010909999, −1.50794620513745349586613851627, 0.53532468186437699340062956663, 1.68262613216661744517272301681, 3.65989087489712313894894252074, 4.26707163289449104376531155645, 5.62071173635456797685048309422, 6.13057936057834348080211189556, 7.10603655704349154770639994242, 7.944490622231826752453659037507, 8.704180586172582240311320712319, 9.993837940225342113099161453989

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