Properties

Label 4-96e4-1.1-c1e2-0-15
Degree $4$
Conductor $84934656$
Sign $1$
Analytic cond. $5415.50$
Root an. cond. $8.57846$
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  − 8·11-s + 8·19-s − 8·25-s + 24·43-s − 14·49-s + 8·59-s + 8·67-s − 20·73-s − 24·83-s − 12·89-s + 16·97-s − 24·107-s − 28·113-s + 26·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 24·169-s + 173-s + 179-s + 181-s + ⋯
L(s)  = 1  − 2.41·11-s + 1.83·19-s − 8/5·25-s + 3.65·43-s − 2·49-s + 1.04·59-s + 0.977·67-s − 2.34·73-s − 2.63·83-s − 1.27·89-s + 1.62·97-s − 2.32·107-s − 2.63·113-s + 2.36·121-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 − 1.84·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 84934656 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84934656 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(84934656\)    =    \(2^{20} \cdot 3^{4}\)
Sign: $1$
Analytic conductor: \(5415.50\)
Root analytic conductor: \(8.57846\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{9216} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 84934656,\ (\ :1/2, 1/2),\ 1)\)

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$C_2$ \( ( 1 + p T^{2} )^{2} \)
11$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
13$C_2^2$ \( 1 + 24 T^{2} + p^{2} T^{4} \)
17$C_2$ \( ( 1 + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 8 T^{2} + p^{2} T^{4} \)
31$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \)
37$C_2^2$ \( 1 + 56 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 104 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
61$C_2^2$ \( 1 - 40 T^{2} + p^{2} T^{4} \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
71$C_2^2$ \( 1 + 110 T^{2} + p^{2} T^{4} \)
73$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
79$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
83$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
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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

−7.57905886980201819024980628196, −7.48342279816428214390378915119, −6.80650694334032464635451875494, −6.72326096792362275148481962425, −5.87776175503213741489436640217, −5.81442414506897741180911285792, −5.41968353479403576294830820611, −5.39115569730116903744311551007, −4.78504891881472247332515221400, −4.45615185450071353202312760467, −3.99938119117503515926209816910, −3.70053432385023940735300044536, −3.03806146398390494923837529719, −2.85845263798419689276635823292, −2.45751339109012390459535941673, −2.17054776791284629784743547097, −1.33585774426506500980667925846, −1.08150098329709409953167514143, 0, 0, 1.08150098329709409953167514143, 1.33585774426506500980667925846, 2.17054776791284629784743547097, 2.45751339109012390459535941673, 2.85845263798419689276635823292, 3.03806146398390494923837529719, 3.70053432385023940735300044536, 3.99938119117503515926209816910, 4.45615185450071353202312760467, 4.78504891881472247332515221400, 5.39115569730116903744311551007, 5.41968353479403576294830820611, 5.81442414506897741180911285792, 5.87776175503213741489436640217, 6.72326096792362275148481962425, 6.80650694334032464635451875494, 7.48342279816428214390378915119, 7.57905886980201819024980628196

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