Properties

Degree 2
Conductor $ 2^{6} \cdot 3^{2} \cdot 7 $
Sign $0.994 + 0.108i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.62 + 2.62i)5-s − 7-s + (0.583 + 0.583i)11-s + (1.76 − 1.76i)13-s + 3.63i·17-s + (−0.963 − 0.963i)19-s − 3.77i·23-s − 8.81i·25-s + (−4.76 − 4.76i)29-s − 4.89i·31-s + (2.62 − 2.62i)35-s + (−4.66 − 4.66i)37-s − 3.83·41-s + (3.45 − 3.45i)43-s + 11.6·47-s + ⋯
L(s)  = 1  + (−1.17 + 1.17i)5-s − 0.377·7-s + (0.175 + 0.175i)11-s + (0.489 − 0.489i)13-s + 0.882i·17-s + (−0.221 − 0.221i)19-s − 0.786i·23-s − 1.76i·25-s + (−0.884 − 0.884i)29-s − 0.878i·31-s + (0.444 − 0.444i)35-s + (−0.767 − 0.767i)37-s − 0.599·41-s + (0.526 − 0.526i)43-s + 1.70·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 + 0.108i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.994 + 0.108i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4032\)    =    \(2^{6} \cdot 3^{2} \cdot 7\)
\( \varepsilon \)  =  $0.994 + 0.108i$
motivic weight  =  \(1\)
character  :  $\chi_{4032} (3599, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 4032,\ (\ :1/2),\ 0.994 + 0.108i)$
$L(1)$  $\approx$  $1.074824413$
$L(\frac12)$  $\approx$  $1.074824413$
$L(\frac{3}{2})$   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 \( 1 + (2.62 - 2.62i)T - 5iT^{2} \)
11 \( 1 + (-0.583 - 0.583i)T + 11iT^{2} \)
13 \( 1 + (-1.76 + 1.76i)T - 13iT^{2} \)
17 \( 1 - 3.63iT - 17T^{2} \)
19 \( 1 + (0.963 + 0.963i)T + 19iT^{2} \)
23 \( 1 + 3.77iT - 23T^{2} \)
29 \( 1 + (4.76 + 4.76i)T + 29iT^{2} \)
31 \( 1 + 4.89iT - 31T^{2} \)
37 \( 1 + (4.66 + 4.66i)T + 37iT^{2} \)
41 \( 1 + 3.83T + 41T^{2} \)
43 \( 1 + (-3.45 + 3.45i)T - 43iT^{2} \)
47 \( 1 - 11.6T + 47T^{2} \)
53 \( 1 + (5.48 - 5.48i)T - 53iT^{2} \)
59 \( 1 + (-4.40 - 4.40i)T + 59iT^{2} \)
61 \( 1 + (7.99 - 7.99i)T - 61iT^{2} \)
67 \( 1 + (-11.3 - 11.3i)T + 67iT^{2} \)
71 \( 1 - 5.85iT - 71T^{2} \)
73 \( 1 + 0.564iT - 73T^{2} \)
79 \( 1 + 16.4iT - 79T^{2} \)
83 \( 1 + (6.92 - 6.92i)T - 83iT^{2} \)
89 \( 1 - 10.7T + 89T^{2} \)
97 \( 1 - 8.64T + 97T^{2} \)
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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

−8.365918083184916474857221136462, −7.52758106993225000140124947586, −7.14787626077241558984923383356, −6.24398390577653769961753991625, −5.70863279881022412361676790548, −4.24927486985077288065121092247, −3.89514817332742471370038961419, −3.03886239705060106875683177470, −2.16675228604298441614341633299, −0.47328970884169858753370038166, 0.72924849450042392686467324211, 1.77098115417686366297851476479, 3.34181931725987524006246046046, 3.74374825851191870341924116674, 4.77365326193491927899738599682, 5.22474594850685538857852291697, 6.30384180694463138711168880712, 7.12241286791810528877602980862, 7.77037298462925773820311176664, 8.523478832515615136413708622699

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