Properties

Degree 2
Conductor $ 2^{4} \cdot 3^{2} \cdot 7 $
Sign $-0.394 + 0.918i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.58 + 0.687i)3-s + (−0.300 − 0.520i)5-s + (−0.5 + 0.866i)7-s + (2.05 − 2.18i)9-s + (−0.800 + 1.38i)11-s + (−0.165 − 0.286i)13-s + (0.834 + 0.620i)15-s − 1.44·17-s − 2.57·19-s + (0.199 − 1.72i)21-s + (0.924 + 1.60i)23-s + (2.31 − 4.01i)25-s + (−1.76 + 4.88i)27-s + (−1.75 + 3.04i)29-s + (−4.81 − 8.33i)31-s + ⋯
L(s)  = 1  + (−0.917 + 0.396i)3-s + (−0.134 − 0.232i)5-s + (−0.188 + 0.327i)7-s + (0.685 − 0.728i)9-s + (−0.241 + 0.417i)11-s + (−0.0458 − 0.0793i)13-s + (0.215 + 0.160i)15-s − 0.351·17-s − 0.591·19-s + (0.0435 − 0.375i)21-s + (0.192 + 0.333i)23-s + (0.463 − 0.803i)25-s + (−0.339 + 0.940i)27-s + (−0.326 + 0.565i)29-s + (−0.863 − 1.49i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.394 + 0.918i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.394 + 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
\( \varepsilon \)  =  $-0.394 + 0.918i$
motivic weight  =  \(1\)
character  :  $\chi_{1008} (673, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 1008,\ (\ :1/2),\ -0.394 + 0.918i)\)
\(L(1)\)  \(\approx\)  \(0.4129193337\)
\(L(\frac12)\)  \(\approx\)  \(0.4129193337\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;7\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;3,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (1.58 - 0.687i)T \)
7 \( 1 + (0.5 - 0.866i)T \)
good5 \( 1 + (0.300 + 0.520i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (0.800 - 1.38i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.165 + 0.286i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 1.44T + 17T^{2} \)
19 \( 1 + 2.57T + 19T^{2} \)
23 \( 1 + (-0.924 - 1.60i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.75 - 3.04i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (4.81 + 8.33i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 0.600T + 37T^{2} \)
41 \( 1 + (3.31 + 5.73i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.81 + 3.14i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-1.95 + 3.38i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 9.27T + 53T^{2} \)
59 \( 1 + (6.93 + 12.0i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.59 + 4.49i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (5.90 + 10.2i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 4.17T + 71T^{2} \)
73 \( 1 - 4.13T + 73T^{2} \)
79 \( 1 + (-4.06 + 7.04i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (2.78 - 4.83i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 3.83T + 89T^{2} \)
97 \( 1 + (-0.974 + 1.68i)T + (-48.5 - 84.0i)T^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−9.685805276123922614225775248096, −9.110334020769514739613741843884, −8.027276898600379143781466527621, −7.01414994334959056649723549654, −6.21530772522529149553718355209, −5.32082342639971756555250664492, −4.54850838360057372906834663418, −3.55892345482151924433836672681, −2.01513183981443905366231857112, −0.21897402390584219998668592611, 1.37982334156225288983331223639, 2.86875618377608157782837725252, 4.16690213423016237680055152272, 5.10452950128811308390306773167, 6.05014117688534965792077859274, 6.82013918375020504130738063845, 7.50748290238223943869866885808, 8.484203434971372320282671041314, 9.492683113747421050501506685805, 10.55153501314154560933218368180

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