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Properties

Label 2-1008-252.223-c1-0-35
Degree $2$
Conductor $1008$
Sign $-0.371 + 0.928i$
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  + (−1.26 − 1.18i)3-s + (0.830 − 0.479i)5-s + (2.56 − 0.657i)7-s + (0.199 + 2.99i)9-s + (−5.27 − 3.04i)11-s + (3.50 − 2.02i)13-s + (−1.61 − 0.376i)15-s − 0.732i·17-s + 3.66·19-s + (−4.01 − 2.20i)21-s + (0.751 − 0.434i)23-s + (−2.03 + 3.53i)25-s + (3.29 − 4.02i)27-s + (2.53 − 4.39i)29-s + (−2.03 − 3.51i)31-s + ⋯
L(s)  = 1  + (−0.730 − 0.683i)3-s + (0.371 − 0.214i)5-s + (0.968 − 0.248i)7-s + (0.0663 + 0.997i)9-s + (−1.59 − 0.918i)11-s + (0.972 − 0.561i)13-s + (−0.417 − 0.0972i)15-s − 0.177i·17-s + 0.840·19-s + (−0.877 − 0.480i)21-s + (0.156 − 0.0905i)23-s + (−0.407 + 0.706i)25-s + (0.633 − 0.773i)27-s + (0.471 − 0.816i)29-s + (−0.364 − 0.631i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.371 + 0.928i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.371 + 0.928i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $-0.371 + 0.928i$
Analytic conductor: \(8.04892\)
Root analytic conductor: \(2.83706\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (223, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1/2),\ -0.371 + 0.928i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.238722470\)
\(L(\frac12)\) \(\approx\) \(1.238722470\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.26 + 1.18i)T \)
7 \( 1 + (-2.56 + 0.657i)T \)
good5 \( 1 + (-0.830 + 0.479i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (5.27 + 3.04i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-3.50 + 2.02i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + 0.732iT - 17T^{2} \)
19 \( 1 - 3.66T + 19T^{2} \)
23 \( 1 + (-0.751 + 0.434i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.53 + 4.39i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (2.03 + 3.51i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 5.58T + 37T^{2} \)
41 \( 1 + (8.17 - 4.72i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (7.69 + 4.44i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-3.68 + 6.37i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 2.07T + 53T^{2} \)
59 \( 1 + (3.10 + 5.37i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (9.20 + 5.31i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.44 - 0.836i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 7.44iT - 71T^{2} \)
73 \( 1 + 7.75iT - 73T^{2} \)
79 \( 1 + (6.68 + 3.85i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-1.17 + 2.03i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 14.5iT - 89T^{2} \)
97 \( 1 + (-11.4 - 6.58i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.965041662078882441905137875007, −8.545933727075617897142057301402, −7.995878567362238796047180451436, −7.33662544860369596439167049181, −6.04152925540021975190562313787, −5.48787077216072360077685171355, −4.78753180502653857813125840259, −3.19873447258056871293255988082, −1.87745308977489674907628702202, −0.64914726098873193086123535899, 1.54848601757038277524278655675, 2.92588000793780275936407218204, 4.29760738468206536989285598533, 5.06606500428878905012383362184, 5.70822320043761571976274152361, 6.74595235924259583333280122396, 7.73120860902926969361288139380, 8.643063678816443951663583715268, 9.555434016955577177535913864656, 10.43310590411329190001753859449

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