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Properties

Label 2-1008-63.41-c1-0-35
Degree $2$
Conductor $1008$
Sign $-0.369 + 0.929i$
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  + (−1.58 + 0.709i)3-s + (1.10 − 1.91i)5-s + (0.906 − 2.48i)7-s + (1.99 − 2.24i)9-s + (2.93 − 1.69i)11-s + (−1.56 − 0.901i)13-s + (−0.388 + 3.80i)15-s − 5.96·17-s + 1.64i·19-s + (0.331 + 4.57i)21-s + (−2.05 − 1.18i)23-s + (0.0556 + 0.0963i)25-s + (−1.56 + 4.95i)27-s + (2.44 − 1.41i)29-s + (−9.28 − 5.36i)31-s + ⋯
L(s)  = 1  + (−0.912 + 0.409i)3-s + (0.494 − 0.856i)5-s + (0.342 − 0.939i)7-s + (0.664 − 0.747i)9-s + (0.885 − 0.511i)11-s + (−0.432 − 0.249i)13-s + (−0.100 + 0.983i)15-s − 1.44·17-s + 0.377i·19-s + (0.0722 + 0.997i)21-s + (−0.428 − 0.247i)23-s + (0.0111 + 0.0192i)25-s + (−0.300 + 0.953i)27-s + (0.453 − 0.262i)29-s + (−1.66 − 0.962i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.369 + 0.929i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.369 + 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $-0.369 + 0.929i$
Analytic conductor: \(8.04892\)
Root analytic conductor: \(2.83706\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (545, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1/2),\ -0.369 + 0.929i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.015462632\)
\(L(\frac12)\) \(\approx\) \(1.015462632\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.58 - 0.709i)T \)
7 \( 1 + (-0.906 + 2.48i)T \)
good5 \( 1 + (-1.10 + 1.91i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-2.93 + 1.69i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.56 + 0.901i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + 5.96T + 17T^{2} \)
19 \( 1 - 1.64iT - 19T^{2} \)
23 \( 1 + (2.05 + 1.18i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.44 + 1.41i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (9.28 + 5.36i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 1.69T + 37T^{2} \)
41 \( 1 + (0.455 - 0.788i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.96 - 3.39i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (0.123 + 0.213i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 7.87iT - 53T^{2} \)
59 \( 1 + (-5.39 + 9.33i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.22 + 0.709i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.99 - 6.91i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 12.1iT - 71T^{2} \)
73 \( 1 - 0.426iT - 73T^{2} \)
79 \( 1 + (2.49 + 4.31i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-4.28 - 7.42i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 10.5T + 89T^{2} \)
97 \( 1 + (6.30 - 3.63i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.661099222107637079897100735013, −9.136102586777498150980089879342, −8.095871750758638129782222948561, −6.97472202293492096659173148945, −6.22497576835007778101706636853, −5.29587808851818735013069005327, −4.47526780011177226300409805189, −3.76851725592972716017202898352, −1.76407065303652448611676691216, −0.51963963339018578131636442208, 1.73925815765465200060985571590, 2.54342501270864060931271799046, 4.22670313461820381444147606043, 5.16916525598109764575312352115, 6.08410204147485688826385697861, 6.77210032448194315494112420855, 7.34735366841283237803209290209, 8.713962462630469565236100493124, 9.393525974592204003284050074328, 10.42729214396634086597804637281

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