Properties

Label 4-4608e2-1.1-c1e2-0-54
Degree $4$
Conductor $21233664$
Sign $1$
Analytic cond. $1353.87$
Root an. cond. $6.06589$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·11-s − 4·17-s − 8·19-s − 8·25-s − 12·41-s − 16·43-s − 12·49-s + 24·59-s − 16·67-s − 16·73-s + 12·83-s − 4·89-s − 28·97-s + 8·107-s − 12·113-s − 10·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 26·169-s + 173-s + ⋯
L(s)  = 1  + 1.20·11-s − 0.970·17-s − 1.83·19-s − 8/5·25-s − 1.87·41-s − 2.43·43-s − 1.71·49-s + 3.12·59-s − 1.95·67-s − 1.87·73-s + 1.31·83-s − 0.423·89-s − 2.84·97-s + 0.773·107-s − 1.12·113-s − 0.909·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 2·169-s + 0.0760·173-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 21233664 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21233664 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(21233664\)    =    \(2^{18} \cdot 3^{4}\)
Sign: $1$
Analytic conductor: \(1353.87\)
Root analytic conductor: \(6.06589\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 21233664,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good5$C_2^2$ \( 1 + 8 T^{2} + p^{2} T^{4} \)
7$C_2^2$ \( 1 + 12 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 + p T^{2} )^{2} \)
17$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - 40 T^{2} + p^{2} T^{4} \)
31$C_2^2$ \( 1 + 12 T^{2} + p^{2} T^{4} \)
37$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 + 86 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 104 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
61$C_2^2$ \( 1 - 78 T^{2} + p^{2} T^{4} \)
67$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
71$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \)
73$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
79$C_2^2$ \( 1 + 140 T^{2} + p^{2} T^{4} \)
83$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \)
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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

−8.145324295612495077388710810711, −8.083400092509309447820919807206, −7.11174278129634174012226832658, −7.09380594818878033884864566768, −6.58216225485557033095722837256, −6.52817398546664431634549984838, −5.99076097101517058868882678531, −5.65796765002052045802713295943, −5.15977761347797516334687653687, −4.69744611970923338206884747787, −4.31680064464861127947847794584, −4.09271876452117206942159769554, −3.50878290635458755194079886524, −3.32264831977791976813729633832, −2.57891094390067338502961656788, −2.01235288403696179765377395125, −1.77329287335247602298611789780, −1.28028641527722770251757283248, 0, 0, 1.28028641527722770251757283248, 1.77329287335247602298611789780, 2.01235288403696179765377395125, 2.57891094390067338502961656788, 3.32264831977791976813729633832, 3.50878290635458755194079886524, 4.09271876452117206942159769554, 4.31680064464861127947847794584, 4.69744611970923338206884747787, 5.15977761347797516334687653687, 5.65796765002052045802713295943, 5.99076097101517058868882678531, 6.52817398546664431634549984838, 6.58216225485557033095722837256, 7.09380594818878033884864566768, 7.11174278129634174012226832658, 8.083400092509309447820919807206, 8.145324295612495077388710810711

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