Properties

Degree 2
Conductor $ 2^{3} \cdot 3^{3} \cdot 7 $
Sign $-0.421 + 0.906i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.23 − 2.13i)5-s + (−0.0665 − 2.64i)7-s + (0.433 + 0.750i)11-s − 1.46·13-s + (−0.933 − 1.61i)17-s + (1.29 − 2.24i)19-s + (0.796 − 1.38i)23-s + (−0.527 − 0.912i)25-s + 5.78·29-s + (−4.78 − 8.28i)31-s + (−5.71 − 3.11i)35-s + (1.66 − 2.87i)37-s + 2.78·41-s − 10.7·43-s + (−6.74 + 11.6i)47-s + ⋯
L(s)  = 1  + (0.550 − 0.952i)5-s + (−0.0251 − 0.999i)7-s + (0.130 + 0.226i)11-s − 0.405·13-s + (−0.226 − 0.392i)17-s + (0.297 − 0.515i)19-s + (0.166 − 0.287i)23-s + (−0.105 − 0.182i)25-s + 1.07·29-s + (−0.859 − 1.48i)31-s + (−0.966 − 0.526i)35-s + (0.272 − 0.472i)37-s + 0.434·41-s − 1.63·43-s + (−0.983 + 1.70i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.421 + 0.906i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.421 + 0.906i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1512\)    =    \(2^{3} \cdot 3^{3} \cdot 7\)
\( \varepsilon \)  =  $-0.421 + 0.906i$
motivic weight  =  \(1\)
character  :  $\chi_{1512} (1297, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 1512,\ (\ :1/2),\ -0.421 + 0.906i)\)
\(L(1)\)  \(\approx\)  \(1.555609437\)
\(L(\frac12)\)  \(\approx\)  \(1.555609437\)
\(L(\frac{3}{2})\)   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 \( 1 \)
7 \( 1 + (0.0665 + 2.64i)T \)
good5 \( 1 + (-1.23 + 2.13i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-0.433 - 0.750i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 1.46T + 13T^{2} \)
17 \( 1 + (0.933 + 1.61i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.29 + 2.24i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.796 + 1.38i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 5.78T + 29T^{2} \)
31 \( 1 + (4.78 + 8.28i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.66 + 2.87i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 2.78T + 41T^{2} \)
43 \( 1 + 10.7T + 43T^{2} \)
47 \( 1 + (6.74 - 11.6i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-3.32 - 5.75i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (3.21 + 5.57i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.09 + 5.35i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.850 + 1.47i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 1.14T + 71T^{2} \)
73 \( 1 + (5.08 + 8.80i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-3.39 + 5.87i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 9.75T + 83T^{2} \)
89 \( 1 + (4.96 - 8.59i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 2.81T + 97T^{2} \)
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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

−9.470081749694597053101964482338, −8.451252100908239204286794038465, −7.60170290646689043118699737325, −6.85129286515118726007100697964, −5.90870072439894434311799229254, −4.84933975492053464254357146098, −4.40652218884129334800096302880, −3.10250844305523989285314385760, −1.78267575912904134625600350754, −0.60557490849267730427019892201, 1.71387728531895547907486711531, 2.72184468797916047512288207293, 3.49306160634205942151300454286, 4.92076073893140022981632368810, 5.68751636107461503494947855598, 6.51806335656454783029936323319, 7.09980761802890769056313522221, 8.328724191576433325072213466430, 8.828102883396627832050092032758, 9.976948743853079486066480990704

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