Properties

Degree 2
Conductor $ 2^{6} \cdot 3^{2} \cdot 7 $
Sign $0.0595 + 0.998i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.96 − 2.96i)5-s − 7-s + (0.569 + 0.569i)11-s + (2.53 − 2.53i)13-s + 1.03i·17-s + (5.23 + 5.23i)19-s − 8.71i·23-s − 12.6i·25-s + (−6.05 − 6.05i)29-s + 3.00i·31-s + (−2.96 + 2.96i)35-s + (−0.149 − 0.149i)37-s + 8.63·41-s + (−1.73 + 1.73i)43-s − 4.10·47-s + ⋯
L(s)  = 1  + (1.32 − 1.32i)5-s − 0.377·7-s + (0.171 + 0.171i)11-s + (0.703 − 0.703i)13-s + 0.251i·17-s + (1.20 + 1.20i)19-s − 1.81i·23-s − 2.52i·25-s + (−1.12 − 1.12i)29-s + 0.539i·31-s + (−0.501 + 0.501i)35-s + (−0.0246 − 0.0246i)37-s + 1.34·41-s + (−0.264 + 0.264i)43-s − 0.599·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0595 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0595 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4032\)    =    \(2^{6} \cdot 3^{2} \cdot 7\)
\( \varepsilon \)  =  $0.0595 + 0.998i$
motivic weight  =  \(1\)
character  :  $\chi_{4032} (3599, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 4032,\ (\ :1/2),\ 0.0595 + 0.998i)$
$L(1)$  $\approx$  $2.479665472$
$L(\frac12)$  $\approx$  $2.479665472$
$L(\frac{3}{2})$   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 \( 1 + (-2.96 + 2.96i)T - 5iT^{2} \)
11 \( 1 + (-0.569 - 0.569i)T + 11iT^{2} \)
13 \( 1 + (-2.53 + 2.53i)T - 13iT^{2} \)
17 \( 1 - 1.03iT - 17T^{2} \)
19 \( 1 + (-5.23 - 5.23i)T + 19iT^{2} \)
23 \( 1 + 8.71iT - 23T^{2} \)
29 \( 1 + (6.05 + 6.05i)T + 29iT^{2} \)
31 \( 1 - 3.00iT - 31T^{2} \)
37 \( 1 + (0.149 + 0.149i)T + 37iT^{2} \)
41 \( 1 - 8.63T + 41T^{2} \)
43 \( 1 + (1.73 - 1.73i)T - 43iT^{2} \)
47 \( 1 + 4.10T + 47T^{2} \)
53 \( 1 + (6.04 - 6.04i)T - 53iT^{2} \)
59 \( 1 + (-6.05 - 6.05i)T + 59iT^{2} \)
61 \( 1 + (-5.81 + 5.81i)T - 61iT^{2} \)
67 \( 1 + (0.0256 + 0.0256i)T + 67iT^{2} \)
71 \( 1 + 14.5iT - 71T^{2} \)
73 \( 1 - 5.38iT - 73T^{2} \)
79 \( 1 - 3.89iT - 79T^{2} \)
83 \( 1 + (-1.40 + 1.40i)T - 83iT^{2} \)
89 \( 1 - 17.0T + 89T^{2} \)
97 \( 1 - 3.75T + 97T^{2} \)
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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

−8.306059196704966634690050651805, −7.78283595770235367676333053420, −6.49238958761718726547533713679, −5.96149745876881219218224803211, −5.42832706234068867695772058175, −4.59408295816790704443371184897, −3.71427883838953405441804139556, −2.56390135653085401397135724491, −1.59835066705095581004627105450, −0.73813084592862037776925907194, 1.34025784625278318802322132160, 2.25873286799453459345344994196, 3.17548094817171199018396452611, 3.71117844406687281639294852727, 5.19580265228937361202741782912, 5.69131659202179536902388068681, 6.49174479524176819595453077091, 7.03609558021018807891280046458, 7.59039233971022537582880070970, 8.929206963656740255158682950200

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