Properties

Degree 2
Conductor $ 2^{6} \cdot 3^{2} \cdot 7 $
Sign $0.816 - 0.577i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.56i·5-s i·7-s − 1.15·11-s + 0.578·13-s + 5.39i·17-s + 6.20i·19-s + 7.62·23-s − 1.57·25-s + 1.41i·29-s + 5.04i·31-s − 2.56·35-s − 9.83·37-s − 6.21i·41-s + 11.2i·43-s − 11.0·47-s + ⋯
L(s)  = 1  − 1.14i·5-s − 0.377i·7-s − 0.346·11-s + 0.160·13-s + 1.30i·17-s + 1.42i·19-s + 1.58·23-s − 0.315·25-s + 0.262i·29-s + 0.906i·31-s − 0.433·35-s − 1.61·37-s − 0.970i·41-s + 1.71i·43-s − 1.61·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.816 - 0.577i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.816 - 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4032\)    =    \(2^{6} \cdot 3^{2} \cdot 7\)
\( \varepsilon \)  =  $0.816 - 0.577i$
motivic weight  =  \(1\)
character  :  $\chi_{4032} (575, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 4032,\ (\ :1/2),\ 0.816 - 0.577i)$
$L(1)$  $\approx$  $1.556913165$
$L(\frac12)$  $\approx$  $1.556913165$
$L(\frac{3}{2})$   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 \( 1 + iT \)
good5 \( 1 + 2.56iT - 5T^{2} \)
11 \( 1 + 1.15T + 11T^{2} \)
13 \( 1 - 0.578T + 13T^{2} \)
17 \( 1 - 5.39iT - 17T^{2} \)
19 \( 1 - 6.20iT - 19T^{2} \)
23 \( 1 - 7.62T + 23T^{2} \)
29 \( 1 - 1.41iT - 29T^{2} \)
31 \( 1 - 5.04iT - 31T^{2} \)
37 \( 1 + 9.83T + 37T^{2} \)
41 \( 1 + 6.21iT - 41T^{2} \)
43 \( 1 - 11.2iT - 43T^{2} \)
47 \( 1 + 11.0T + 47T^{2} \)
53 \( 1 - 4.53iT - 53T^{2} \)
59 \( 1 - 4.83T + 59T^{2} \)
61 \( 1 + 0.951T + 61T^{2} \)
67 \( 1 + 2.78iT - 67T^{2} \)
71 \( 1 - 3.68T + 71T^{2} \)
73 \( 1 - 14.0T + 73T^{2} \)
79 \( 1 - 12.8iT - 79T^{2} \)
83 \( 1 - 8.77T + 83T^{2} \)
89 \( 1 - 5.68iT - 89T^{2} \)
97 \( 1 + 12.8T + 97T^{2} \)
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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

−8.389512156897243121261599821040, −8.069046817809705471820544099422, −7.04021333504557652311742856131, −6.30459908708859962003581518211, −5.33927123085062869699311475463, −4.90656352349358862332723731601, −3.91747364297512394945235982032, −3.23605205487389748954127461677, −1.76824760168611945423313129837, −1.07532066234199239964883932544, 0.50113396288982743828237085438, 2.14326820126507998570725508745, 2.89077657299465332597075922509, 3.45790715269797027894047300192, 4.83345936603194771832575835671, 5.23296238309964372868419066909, 6.37496189731986800320986297257, 6.97339208594911334774738698233, 7.37365576788897054138911530296, 8.431452202451274676153990428566

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