Properties

Degree 2
Conductor $ 2^{4} \cdot 3 \cdot 5 \cdot 7^{2} \cdot 13 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 2

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 5-s + 9-s − 13-s − 15-s − 6·17-s − 4·19-s + 25-s + 27-s + 6·29-s − 4·31-s − 10·37-s − 39-s − 6·41-s − 8·43-s − 45-s − 6·51-s − 6·53-s − 4·57-s − 12·59-s − 14·61-s + 65-s + 4·67-s − 2·73-s + 75-s − 8·79-s + 81-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.447·5-s + 1/3·9-s − 0.277·13-s − 0.258·15-s − 1.45·17-s − 0.917·19-s + 1/5·25-s + 0.192·27-s + 1.11·29-s − 0.718·31-s − 1.64·37-s − 0.160·39-s − 0.937·41-s − 1.21·43-s − 0.149·45-s − 0.840·51-s − 0.824·53-s − 0.529·57-s − 1.56·59-s − 1.79·61-s + 0.124·65-s + 0.488·67-s − 0.234·73-s + 0.115·75-s − 0.900·79-s + 1/9·81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 152880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 152880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(152880\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 7^{2} \cdot 13\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{152880} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  2
Selberg data  =  $(2,\ 152880,\ (\ :1/2),\ 1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5,\;7,\;13\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;5,\;7,\;13\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - T \)
5 \( 1 + T \)
7 \( 1 \)
13 \( 1 + T \)
good11 \( 1 + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−13.77790482932362, −13.42856529533599, −12.71205520309013, −12.55772037669632, −11.89255840515740, −11.47212032704420, −10.81748871152037, −10.52631434653771, −10.02729166462434, −9.330673752109021, −8.923235784658411, −8.458813372784856, −8.175716857317191, −7.437343464308774, −6.998163392381160, −6.519718207958418, −6.123260269903604, −5.101434919459077, −4.858143233173764, −4.199855537505329, −3.784515163655208, −3.008581092400540, −2.674331241128836, −1.766117770964027, −1.494657291872529, 0, 0, 1.494657291872529, 1.766117770964027, 2.674331241128836, 3.008581092400540, 3.784515163655208, 4.199855537505329, 4.858143233173764, 5.101434919459077, 6.123260269903604, 6.519718207958418, 6.998163392381160, 7.437343464308774, 8.175716857317191, 8.458813372784856, 8.923235784658411, 9.330673752109021, 10.02729166462434, 10.52631434653771, 10.81748871152037, 11.47212032704420, 11.89255840515740, 12.55772037669632, 12.71205520309013, 13.42856529533599, 13.77790482932362

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