Properties

Degree 2
Conductor $ 2^{6} \cdot 3^{2} \cdot 7 $
Sign $-0.417 - 0.908i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.417 - 0.908i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.417 - 0.908i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4032\)    =    \(2^{6} \cdot 3^{2} \cdot 7\)
\( \varepsilon \)  =  $-0.417 - 0.908i$
motivic weight  =  \(1\)
character  :  $\chi_{4032} (3599, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 4032,\ (\ :1/2),\ -0.417 - 0.908i)$
$L(1)$  $\approx$  $1.097697352$
$L(\frac12)$  $\approx$  $1.097697352$
$L(\frac{3}{2})$   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

−8.703677991825013504911564430843, −7.78236322160435729889830105464, −7.33038063638632771137248481569, −6.40931165747377994361529072165, −5.97296369873004281842333645455, −4.81760478539520796601869093086, −3.94863763989858136991493618883, −3.32329317624028099415861503859, −2.48334420805933749804018656527, −1.10160626949482344769297383339, 0.36835501851288102039301171742, 1.45806762928041246259225830986, 2.73118433618980860955347271339, 3.76151758725359021621303743991, 4.29660689256743173541367549963, 5.11265075424814277895089051361, 6.12231307248142144448581387870, 6.64669305287690372370869646405, 7.61855224847689593719114793847, 8.251525131939462300682962216450

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