Properties

Degree 2
Conductor $ 2^{3} \cdot 3^{3} \cdot 7 $
Sign $0.472 + 0.881i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.497 + 1.32i)2-s + (−1.50 − 1.31i)4-s − 1.25i·5-s + 7-s + (2.49 − 1.33i)8-s + (1.65 + 0.624i)10-s − 1.55i·11-s − 1.07i·13-s + (−0.497 + 1.32i)14-s + (0.528 + 3.96i)16-s − 0.0158·17-s − 2.35i·19-s + (−1.65 + 1.88i)20-s + (2.06 + 0.775i)22-s − 5.95·23-s + ⋯
L(s)  = 1  + (−0.351 + 0.936i)2-s + (−0.752 − 0.658i)4-s − 0.560i·5-s + 0.377·7-s + (0.881 − 0.472i)8-s + (0.524 + 0.197i)10-s − 0.469i·11-s − 0.297i·13-s + (−0.133 + 0.353i)14-s + (0.132 + 0.991i)16-s − 0.00383·17-s − 0.539i·19-s + (−0.369 + 0.421i)20-s + (0.439 + 0.165i)22-s − 1.24·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.472 + 0.881i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.472 + 0.881i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1512\)    =    \(2^{3} \cdot 3^{3} \cdot 7\)
\( \varepsilon \)  =  $0.472 + 0.881i$
motivic weight  =  \(1\)
character  :  $\chi_{1512} (757, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 1512,\ (\ :1/2),\ 0.472 + 0.881i)$
$L(1)$  $\approx$  $0.9262378755$
$L(\frac12)$  $\approx$  $0.9262378755$
$L(\frac{3}{2})$   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 \( 1 + (0.497 - 1.32i)T \)
3 \( 1 \)
7 \( 1 - T \)
good5 \( 1 + 1.25iT - 5T^{2} \)
11 \( 1 + 1.55iT - 11T^{2} \)
13 \( 1 + 1.07iT - 13T^{2} \)
17 \( 1 + 0.0158T + 17T^{2} \)
19 \( 1 + 2.35iT - 19T^{2} \)
23 \( 1 + 5.95T + 23T^{2} \)
29 \( 1 + 0.469iT - 29T^{2} \)
31 \( 1 - 1.69T + 31T^{2} \)
37 \( 1 + 4.59iT - 37T^{2} \)
41 \( 1 + 12.2T + 41T^{2} \)
43 \( 1 - 1.97iT - 43T^{2} \)
47 \( 1 + 7.12T + 47T^{2} \)
53 \( 1 + 1.86iT - 53T^{2} \)
59 \( 1 + 8.54iT - 59T^{2} \)
61 \( 1 + 3.92iT - 61T^{2} \)
67 \( 1 + 12.7iT - 67T^{2} \)
71 \( 1 - 4.22T + 71T^{2} \)
73 \( 1 - 6.51T + 73T^{2} \)
79 \( 1 + 6.15T + 79T^{2} \)
83 \( 1 + 8.88iT - 83T^{2} \)
89 \( 1 - 0.240T + 89T^{2} \)
97 \( 1 - 5.86T + 97T^{2} \)
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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

−9.151361643928831778720644613143, −8.347526728974102883201278372973, −7.965326563931488582968330520096, −6.89881278875915729812455253086, −6.14615048623035564249343042303, −5.21611833943150813739512904355, −4.63635193974363990279618087419, −3.47709932208345114257042253059, −1.78639506443631589969422330609, −0.42994867035734544951495645548, 1.45131256870593141120898723001, 2.42932936149613515478674991721, 3.47034965893330389781848324820, 4.35920244947254011644787919560, 5.25690169770811135668024984162, 6.50076068970241366338445690151, 7.36861088077420881098758933547, 8.212307669713270102365119755433, 8.828980792225812841110494640152, 10.03396515344661784197045079156

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