Properties

Label 4-96e4-1.1-c1e2-0-19
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

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Normalization:  

Dirichlet series

L(s)  = 1  + 12·17-s − 12·19-s − 10·25-s + 12·41-s − 12·43-s − 12·49-s + 8·67-s − 24·83-s − 12·89-s − 24·97-s − 24·107-s + 12·113-s − 22·121-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 + 191-s + ⋯
L(s)  = 1  + 2.91·17-s − 2.75·19-s − 2·25-s + 1.87·41-s − 1.82·43-s − 1.71·49-s + 0.977·67-s − 2.63·83-s − 1.27·89-s − 2.43·97-s − 2.32·107-s + 1.12·113-s − 2·121-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 + 0.0723·191-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$C_2$ \( ( 1 + p T^{2} )^{2} \)
7$C_2^2$ \( 1 + 12 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2^2$ \( 1 + 24 T^{2} + p^{2} T^{4} \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 26 T^{2} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \)
31$C_2^2$ \( 1 + 60 T^{2} + p^{2} T^{4} \)
37$C_2^2$ \( 1 + 24 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 + 22 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 34 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2^2$ \( 1 + 72 T^{2} + p^{2} T^{4} \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
71$C_2^2$ \( 1 + 70 T^{2} + p^{2} T^{4} \)
73$C_2$ \( ( 1 + p T^{2} )^{2} \)
79$C_2^2$ \( 1 + 156 T^{2} + p^{2} T^{4} \)
83$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
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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

−7.51570025718327390381176918879, −7.25943945681301135771745750159, −6.92729673479926931669830924921, −6.36975596317611075848566902319, −6.23592645849099751279148135520, −5.81841173945510348153226364486, −5.45229323191015074898644921468, −5.38968816664649620041112746909, −4.63684437840702084272916085732, −4.40522038893178953404648831657, −3.91822001331664182388191006991, −3.81342148689937479664976276627, −3.10432066055094047533911296844, −3.04516474317418007342515769761, −2.23942291650490898635845795027, −2.08706444672283720155282572544, −1.33845782706610628471002971403, −1.21087440679347809127984064620, 0, 0, 1.21087440679347809127984064620, 1.33845782706610628471002971403, 2.08706444672283720155282572544, 2.23942291650490898635845795027, 3.04516474317418007342515769761, 3.10432066055094047533911296844, 3.81342148689937479664976276627, 3.91822001331664182388191006991, 4.40522038893178953404648831657, 4.63684437840702084272916085732, 5.38968816664649620041112746909, 5.45229323191015074898644921468, 5.81841173945510348153226364486, 6.23592645849099751279148135520, 6.36975596317611075848566902319, 6.92729673479926931669830924921, 7.25943945681301135771745750159, 7.51570025718327390381176918879

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