Properties

Degree 2
Conductor $ 2^{6} \cdot 5 \cdot 11^{2} $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5-s + 4·7-s − 3·9-s − 2·13-s − 2·17-s + 4·19-s + 4·23-s + 25-s − 2·29-s − 8·31-s − 4·35-s − 6·37-s + 6·41-s − 8·43-s + 3·45-s + 4·47-s + 9·49-s − 6·53-s + 4·59-s − 2·61-s − 12·63-s + 2·65-s − 8·67-s + 6·73-s + 9·81-s − 16·83-s + 2·85-s + ⋯
L(s)  = 1  − 0.447·5-s + 1.51·7-s − 9-s − 0.554·13-s − 0.485·17-s + 0.917·19-s + 0.834·23-s + 1/5·25-s − 0.371·29-s − 1.43·31-s − 0.676·35-s − 0.986·37-s + 0.937·41-s − 1.21·43-s + 0.447·45-s + 0.583·47-s + 9/7·49-s − 0.824·53-s + 0.520·59-s − 0.256·61-s − 1.51·63-s + 0.248·65-s − 0.977·67-s + 0.702·73-s + 81-s − 1.75·83-s + 0.216·85-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 38720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(38720\)    =    \(2^{6} \cdot 5 \cdot 11^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{38720} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 38720,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.721728668$
$L(\frac12)$  $\approx$  $1.721728668$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;5,\;11\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;5,\;11\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + T \)
11 \( 1 \)
good3 \( 1 + p T^{2} \)
7 \( 1 - 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 16 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−14.70630789827023, −14.46294565835162, −13.86897077473220, −13.42523074077869, −12.53029955065840, −12.24207292327578, −11.40517545113363, −11.28627578586871, −10.91665130366986, −10.12631248975805, −9.397026201387884, −8.819666963823538, −8.467099818468439, −7.834791558464124, −7.309376412262630, −6.939488081306537, −5.859092659199151, −5.434209418927068, −4.887577573494345, −4.402904405334209, −3.514003304899559, −2.963312831087347, −2.109813912594614, −1.500729690642643, −0.4804444177767213, 0.4804444177767213, 1.500729690642643, 2.109813912594614, 2.963312831087347, 3.514003304899559, 4.402904405334209, 4.887577573494345, 5.434209418927068, 5.859092659199151, 6.939488081306537, 7.309376412262630, 7.834791558464124, 8.467099818468439, 8.819666963823538, 9.397026201387884, 10.12631248975805, 10.91665130366986, 11.28627578586871, 11.40517545113363, 12.24207292327578, 12.53029955065840, 13.42523074077869, 13.86897077473220, 14.46294565835162, 14.70630789827023

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