Properties

Degree 2
Conductor $ 2^{6} \cdot 3^{2} \cdot 7 $
Sign $-0.717 - 0.696i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.65 + 1.65i)5-s − 7-s + (−0.993 + 0.993i)11-s + (2.62 + 2.62i)13-s + 2.77i·17-s + (−1.56 + 1.56i)19-s − 1.05i·23-s + 0.491i·25-s + (−3.47 + 3.47i)29-s + 1.06i·31-s + (−1.65 − 1.65i)35-s + (−0.0657 + 0.0657i)37-s − 6.31·41-s + (2.38 + 2.38i)43-s − 1.47·47-s + ⋯
L(s)  = 1  + (0.741 + 0.741i)5-s − 0.377·7-s + (−0.299 + 0.299i)11-s + (0.728 + 0.728i)13-s + 0.673i·17-s + (−0.358 + 0.358i)19-s − 0.219i·23-s + 0.0982i·25-s + (−0.645 + 0.645i)29-s + 0.190i·31-s + (−0.280 − 0.280i)35-s + (−0.0108 + 0.0108i)37-s − 0.985·41-s + (0.364 + 0.364i)43-s − 0.214·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.717 - 0.696i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.717 - 0.696i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4032\)    =    \(2^{6} \cdot 3^{2} \cdot 7\)
\( \varepsilon \)  =  $-0.717 - 0.696i$
motivic weight  =  \(1\)
character  :  $\chi_{4032} (1583, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 4032,\ (\ :1/2),\ -0.717 - 0.696i)$
$L(1)$  $\approx$  $1.404674316$
$L(\frac12)$  $\approx$  $1.404674316$
$L(\frac{3}{2})$   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 \( 1 + T \)
good5 \( 1 + (-1.65 - 1.65i)T + 5iT^{2} \)
11 \( 1 + (0.993 - 0.993i)T - 11iT^{2} \)
13 \( 1 + (-2.62 - 2.62i)T + 13iT^{2} \)
17 \( 1 - 2.77iT - 17T^{2} \)
19 \( 1 + (1.56 - 1.56i)T - 19iT^{2} \)
23 \( 1 + 1.05iT - 23T^{2} \)
29 \( 1 + (3.47 - 3.47i)T - 29iT^{2} \)
31 \( 1 - 1.06iT - 31T^{2} \)
37 \( 1 + (0.0657 - 0.0657i)T - 37iT^{2} \)
41 \( 1 + 6.31T + 41T^{2} \)
43 \( 1 + (-2.38 - 2.38i)T + 43iT^{2} \)
47 \( 1 + 1.47T + 47T^{2} \)
53 \( 1 + (-7.63 - 7.63i)T + 53iT^{2} \)
59 \( 1 + (4.15 - 4.15i)T - 59iT^{2} \)
61 \( 1 + (7.78 + 7.78i)T + 61iT^{2} \)
67 \( 1 + (1.98 - 1.98i)T - 67iT^{2} \)
71 \( 1 + 13.0iT - 71T^{2} \)
73 \( 1 + 9.50iT - 73T^{2} \)
79 \( 1 - 9.85iT - 79T^{2} \)
83 \( 1 + (-1.13 - 1.13i)T + 83iT^{2} \)
89 \( 1 + 7.04T + 89T^{2} \)
97 \( 1 - 5.35T + 97T^{2} \)
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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

−8.847702869112784531978719333117, −7.956255658529928597910527181976, −7.13820764440150482564899379575, −6.34847438452488549620784510907, −6.06840498316618870284943607846, −5.03313296254870035107094108523, −4.05772522590265725506838375011, −3.27105020908619148892613646080, −2.30497765930989401547153806697, −1.49756696195347056290504230550, 0.38431836768571028454798239769, 1.50617121741805089086650618208, 2.58363325280024470568732950231, 3.46715483715206153871686312412, 4.42853934597530084830170912839, 5.43362085222428549493167822614, 5.70631132137435664322787825657, 6.63974439577708854312654955243, 7.44887374654742652649007259479, 8.340024727182592496724007660883

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