Properties

Degree 2
Conductor $ 2^{3} \cdot 3^{3} \cdot 7 $
Sign $0.444 - 0.895i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.885 − 1.10i)2-s + (−0.430 − 1.95i)4-s + 3.50i·5-s − 7-s + (−2.53 − 1.25i)8-s + (3.85 + 3.10i)10-s − 3.01i·11-s + 3.90i·13-s + (−0.885 + 1.10i)14-s + (−3.62 + 1.68i)16-s + 1.38·17-s + 4.79i·19-s + (6.83 − 1.50i)20-s + (−3.32 − 2.66i)22-s − 5.06·23-s + ⋯
L(s)  = 1  + (0.626 − 0.779i)2-s + (−0.215 − 0.976i)4-s + 1.56i·5-s − 0.377·7-s + (−0.895 − 0.444i)8-s + (1.22 + 0.980i)10-s − 0.908i·11-s + 1.08i·13-s + (−0.236 + 0.294i)14-s + (−0.907 + 0.420i)16-s + 0.336·17-s + 1.09i·19-s + (1.52 − 0.336i)20-s + (−0.708 − 0.569i)22-s − 1.05·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.444 - 0.895i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.444 - 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1512\)    =    \(2^{3} \cdot 3^{3} \cdot 7\)
\( \varepsilon \)  =  $0.444 - 0.895i$
motivic weight  =  \(1\)
character  :  $\chi_{1512} (757, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 1512,\ (\ :1/2),\ 0.444 - 0.895i)$
$L(1)$  $\approx$  $1.447495971$
$L(\frac12)$  $\approx$  $1.447495971$
$L(\frac{3}{2})$   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 \( 1 + (-0.885 + 1.10i)T \)
3 \( 1 \)
7 \( 1 + T \)
good5 \( 1 - 3.50iT - 5T^{2} \)
11 \( 1 + 3.01iT - 11T^{2} \)
13 \( 1 - 3.90iT - 13T^{2} \)
17 \( 1 - 1.38T + 17T^{2} \)
19 \( 1 - 4.79iT - 19T^{2} \)
23 \( 1 + 5.06T + 23T^{2} \)
29 \( 1 - 4.91iT - 29T^{2} \)
31 \( 1 - 1.13T + 31T^{2} \)
37 \( 1 - 9.45iT - 37T^{2} \)
41 \( 1 - 4.11T + 41T^{2} \)
43 \( 1 + 1.51iT - 43T^{2} \)
47 \( 1 + 10.7T + 47T^{2} \)
53 \( 1 - 0.431iT - 53T^{2} \)
59 \( 1 - 7.40iT - 59T^{2} \)
61 \( 1 - 12.9iT - 61T^{2} \)
67 \( 1 - 3.36iT - 67T^{2} \)
71 \( 1 + 6.26T + 71T^{2} \)
73 \( 1 - 10.0T + 73T^{2} \)
79 \( 1 - 12.9T + 79T^{2} \)
83 \( 1 + 17.4iT - 83T^{2} \)
89 \( 1 + 0.818T + 89T^{2} \)
97 \( 1 + 11.4T + 97T^{2} \)
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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

−10.04044480869421376865855508186, −9.064455421736245692455394623387, −8.001444464149014382498158101258, −6.84561305348279814524393944439, −6.31037690092101873187576529373, −5.61271802019659160333987289115, −4.24400318126805454484753243623, −3.43902347973611141403782929796, −2.80682387758572779910389643509, −1.63164554247219376232966888506, 0.44763152747786232055372405587, 2.25904216981508768885641649535, 3.61049283012097217116922153020, 4.52654407659850902230654783679, 5.14439054523114394930359089045, 5.88304255666201946116784818123, 6.83710239824075736579339947070, 7.944990579334591157463447004285, 8.165148670902142639328747756614, 9.389128719527514221599043254787

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