Properties

Degree 2
Conductor $ 2^{5} \cdot 3^{2} \cdot 7 $
Sign $-i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.41i·5-s i·7-s − 1.41·11-s + 1.41i·17-s + 2i·19-s + 1.41·23-s − 1.00·25-s + 1.41·35-s + 1.41i·41-s − 49-s − 2.00i·55-s − 1.41·71-s + 1.41i·77-s − 2.00·85-s − 1.41i·89-s + ⋯
L(s)  = 1  + 1.41i·5-s i·7-s − 1.41·11-s + 1.41i·17-s + 2i·19-s + 1.41·23-s − 1.00·25-s + 1.41·35-s + 1.41i·41-s − 49-s − 2.00i·55-s − 1.41·71-s + 1.41i·77-s − 2.00·85-s − 1.41i·89-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(2016\)    =    \(2^{5} \cdot 3^{2} \cdot 7\)
\( \varepsilon \)  =  $-i$
motivic weight  =  \(0\)
character  :  $\chi_{2016} (1441, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 2016,\ (\ :0),\ -i)$
$L(\frac{1}{2})$  $\approx$  $0.9525154182$
$L(\frac12)$  $\approx$  $0.9525154182$
$L(1)$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;7\}$, \(F_p\) is a polynomial of degree 2. If $p \in \{2,\;3,\;7\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 + iT \)
good5 \( 1 - 1.41iT - T^{2} \)
11 \( 1 + 1.41T + T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 - 1.41iT - T^{2} \)
19 \( 1 - 2iT - T^{2} \)
23 \( 1 - 1.41T + T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 - 1.41iT - T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 + 1.41T + T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + 1.41iT - T^{2} \)
97 \( 1 - T^{2} \)
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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

−10.07116775517939938096824052210, −8.545693792569315061123293041102, −7.77572705966048293672151531354, −7.32714659515117966369010023572, −6.39954639714155647403833438354, −5.75967228228394173153998636269, −4.57565632907693409972007276983, −3.55734639438890375042461572094, −2.95097373307377632743712870248, −1.66575440404263102548913708498, 0.68315849343281942353411925020, 2.32073328036948585527311670018, 3.01658380050073067093677843413, 4.72460924295937248678957118743, 5.01696770359342235398471554338, 5.61589766192628420569618602160, 6.91614135001548020374024290536, 7.64152648503015109272804164024, 8.680170265363106022567026767162, 8.976154012205755560166354338268

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