Properties

Degree 2
Conductor $ 2^{4} \cdot 3 \cdot 5 \cdot 7^{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  + 3-s − 5-s + 9-s + 4·11-s + 2·13-s − 15-s − 2·17-s + 4·19-s + 8·23-s + 25-s + 27-s − 2·29-s + 4·33-s + 6·37-s + 2·39-s + 6·41-s + 4·43-s − 45-s − 2·51-s − 10·53-s − 4·55-s + 4·57-s + 12·59-s − 14·61-s − 2·65-s + 12·67-s + 8·69-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.447·5-s + 1/3·9-s + 1.20·11-s + 0.554·13-s − 0.258·15-s − 0.485·17-s + 0.917·19-s + 1.66·23-s + 1/5·25-s + 0.192·27-s − 0.371·29-s + 0.696·33-s + 0.986·37-s + 0.320·39-s + 0.937·41-s + 0.609·43-s − 0.149·45-s − 0.280·51-s − 1.37·53-s − 0.539·55-s + 0.529·57-s + 1.56·59-s − 1.79·61-s − 0.248·65-s + 1.46·67-s + 0.963·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 11760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(11760\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 7^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{11760} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 11760,\ (\ :1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(3.102129835\)
\(L(\frac12)\)  \(\approx\)  \(3.102129835\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5,\;7\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;5,\;7\}$, 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 \( 1 \)
good11 \( 1 - 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
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

−16.26158625921562, −15.84543601539716, −15.20294581811562, −14.70849924015988, −14.18203728292491, −13.67135354623125, −12.89587441223978, −12.59721548377167, −11.68881647798720, −11.19813910461926, −10.87317498589497, −9.677902214792115, −9.466387637789692, −8.735618317591577, −8.313139190968788, −7.371951328715601, −7.070494082123913, −6.274411071276473, −5.554190358748549, −4.575819234590486, −4.097742224731950, −3.307662455264993, −2.731199437603262, −1.555455497411807, −0.8532442649986090, 0.8532442649986090, 1.555455497411807, 2.731199437603262, 3.307662455264993, 4.097742224731950, 4.575819234590486, 5.554190358748549, 6.274411071276473, 7.070494082123913, 7.371951328715601, 8.313139190968788, 8.735618317591577, 9.466387637789692, 9.677902214792115, 10.87317498589497, 11.19813910461926, 11.68881647798720, 12.59721548377167, 12.89587441223978, 13.67135354623125, 14.18203728292491, 14.70849924015988, 15.20294581811562, 15.84543601539716, 16.26158625921562

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