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Properties

Label 2-1008-9.7-c1-0-23
Degree $2$
Conductor $1008$
Sign $0.904 - 0.426i$
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  + (1.73 − 0.0789i)3-s + (1.29 + 2.24i)5-s + (−0.5 + 0.866i)7-s + (2.98 − 0.273i)9-s + (2.25 − 3.90i)11-s + (−0.5 − 0.866i)13-s + (2.42 + 3.78i)15-s − 0.945·17-s + 4.05·19-s + (−0.796 + 1.53i)21-s + (−0.136 − 0.236i)23-s + (−0.863 + 1.49i)25-s + (5.14 − 0.708i)27-s + (−1.23 + 2.13i)29-s + (1.16 + 2.01i)31-s + ⋯
L(s)  = 1  + (0.998 − 0.0455i)3-s + (0.579 + 1.00i)5-s + (−0.188 + 0.327i)7-s + (0.995 − 0.0910i)9-s + (0.680 − 1.17i)11-s + (−0.138 − 0.240i)13-s + (0.625 + 0.977i)15-s − 0.229·17-s + 0.930·19-s + (−0.173 + 0.335i)21-s + (−0.0284 − 0.0493i)23-s + (−0.172 + 0.299i)25-s + (0.990 − 0.136i)27-s + (−0.228 + 0.395i)29-s + (0.209 + 0.362i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.904 - 0.426i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.904 - 0.426i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $0.904 - 0.426i$
Analytic conductor: \(8.04892\)
Root analytic conductor: \(2.83706\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (673, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1/2),\ 0.904 - 0.426i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.611729373\)
\(L(\frac12)\) \(\approx\) \(2.611729373\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.73 + 0.0789i)T \)
7 \( 1 + (0.5 - 0.866i)T \)
good5 \( 1 + (-1.29 - 2.24i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-2.25 + 3.90i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.5 + 0.866i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 0.945T + 17T^{2} \)
19 \( 1 - 4.05T + 19T^{2} \)
23 \( 1 + (0.136 + 0.236i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.23 - 2.13i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-1.16 - 2.01i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 1.78T + 37T^{2} \)
41 \( 1 + (-3.20 - 5.54i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (5.21 - 9.03i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (6.08 - 10.5i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 6.27T + 53T^{2} \)
59 \( 1 + (1.36 + 2.36i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.13 + 1.96i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (7.90 + 13.6i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 3.27T + 71T^{2} \)
73 \( 1 + 1.50T + 73T^{2} \)
79 \( 1 + (-7.35 + 12.7i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (0.472 - 0.819i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 14.3T + 89T^{2} \)
97 \( 1 + (-5.74 + 9.95i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.767960326116008218474848850791, −9.335374448846201344077257991875, −8.379528300720731602747199929502, −7.60218270354300100485209949262, −6.56319432148311317926475824781, −6.04823882979883046257686687473, −4.65730624942320504115911071673, −3.19964595470058327890713143638, −2.97943582233612922475835676594, −1.52231272305937092667384380036, 1.33067511568237591796028561530, 2.27743933639922358566135441405, 3.71041576967219043348630829496, 4.51209605173735546994510245801, 5.41211936788369667879946622538, 6.76939163158079384310928014019, 7.40394866595141670185213091073, 8.418010802124935901368253940144, 9.212223250489388875720483049935, 9.653454211580814695207240883883

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