Properties

Degree $4$
Conductor $9216$
Sign $1$
Motivic weight $1$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s + 3·9-s − 8·11-s + 4·17-s + 8·19-s − 6·25-s + 4·27-s − 16·33-s − 12·41-s − 8·43-s − 14·49-s + 8·51-s + 16·57-s − 8·59-s + 8·67-s + 20·73-s − 12·75-s + 5·81-s + 8·83-s − 12·89-s + 4·97-s − 24·99-s + 24·107-s + 36·113-s + 26·121-s − 24·123-s + 127-s + ⋯
L(s)  = 1  + 1.15·3-s + 9-s − 2.41·11-s + 0.970·17-s + 1.83·19-s − 6/5·25-s + 0.769·27-s − 2.78·33-s − 1.87·41-s − 1.21·43-s − 2·49-s + 1.12·51-s + 2.11·57-s − 1.04·59-s + 0.977·67-s + 2.34·73-s − 1.38·75-s + 5/9·81-s + 0.878·83-s − 1.27·89-s + 0.406·97-s − 2.41·99-s + 2.32·107-s + 3.38·113-s + 2.36·121-s − 2.16·123-s + 0.0887·127-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9216 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9216 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(9216\)    =    \(2^{10} \cdot 3^{2}\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{9216} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 9216,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.285289264\)
\(L(\frac12)\) \(\approx\) \(1.285289264\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad2 \( 1 \)
3$C_1$ \( ( 1 - T )^{2} \)
good5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
7$C_2$ \( ( 1 + p T^{2} )^{2} \)
11$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
59$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
73$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.58676923620628310825913776398, −11.02059068705223596291987853205, −10.20677110069933956172337352464, −9.785656483951954927896383611288, −9.656797755147918818405236990829, −8.456614023795656741730916500766, −8.173282349834764667604075089133, −7.61947548271568794045481490015, −7.26958585488936859387169650645, −6.16826675668887144123811676692, −5.14444399128618341481533912587, −5.01357758335939619070346679818, −3.43299575130251299450107440988, −3.14826068230388029805668446658, −1.99999348241168201324552330088, 1.99999348241168201324552330088, 3.14826068230388029805668446658, 3.43299575130251299450107440988, 5.01357758335939619070346679818, 5.14444399128618341481533912587, 6.16826675668887144123811676692, 7.26958585488936859387169650645, 7.61947548271568794045481490015, 8.173282349834764667604075089133, 8.456614023795656741730916500766, 9.656797755147918818405236990829, 9.785656483951954927896383611288, 10.20677110069933956172337352464, 11.02059068705223596291987853205, 11.58676923620628310825913776398

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