Properties

Label 4-96e4-1.1-c1e2-0-10
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 $0$

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: \(0\)
Selberg data: \((4,\ 84934656,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.659282494\)
\(L(\frac12)\) \(\approx\) \(4.659282494\)
\(L(\frac{3}{2})\) 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

−7.84427175305299608550158746036, −7.46043798629235467263822737250, −7.40167465717708549279967444845, −7.15918171128034451875335485390, −6.28679465402374556932422389631, −6.17385746999357038128225417477, −5.68997360923086839170326729152, −5.62803513112500294766193716965, −5.16513294336308053359983400557, −4.98200223068025001004779853322, −4.31759283500191200793124269424, −3.91656388657350585020581216377, −3.63725645635764907886784885763, −3.24840560741150090191617676137, −2.86254695423894142971422657277, −2.63183069777450192893620479739, −1.76216649251033252506084695380, −1.49352735950654379460316768828, −0.913083715822749780684548732571, −0.61295183932079142230398385966, 0.61295183932079142230398385966, 0.913083715822749780684548732571, 1.49352735950654379460316768828, 1.76216649251033252506084695380, 2.63183069777450192893620479739, 2.86254695423894142971422657277, 3.24840560741150090191617676137, 3.63725645635764907886784885763, 3.91656388657350585020581216377, 4.31759283500191200793124269424, 4.98200223068025001004779853322, 5.16513294336308053359983400557, 5.62803513112500294766193716965, 5.68997360923086839170326729152, 6.17385746999357038128225417477, 6.28679465402374556932422389631, 7.15918171128034451875335485390, 7.40167465717708549279967444845, 7.46043798629235467263822737250, 7.84427175305299608550158746036

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