Properties

Label 4-96e4-1.1-c1e2-0-2
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

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 8·17-s + 16·19-s − 8·25-s + 8·41-s − 6·49-s − 24·59-s − 8·67-s + 28·73-s − 32·83-s − 12·89-s + 32·97-s − 24·107-s + 36·113-s − 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 8·169-s + 173-s + 179-s + 181-s + ⋯
L(s)  = 1  − 1.94·17-s + 3.67·19-s − 8/5·25-s + 1.24·41-s − 6/7·49-s − 3.12·59-s − 0.977·67-s + 3.27·73-s − 3.51·83-s − 1.27·89-s + 3.24·97-s − 2.32·107-s + 3.38·113-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 − 0.615·169-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: \(0\)
Selberg data: \((4,\ 84934656,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.061204393\)
\(L(\frac12)\) \(\approx\) \(2.061204393\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2^2$ \( 1 + 8 T^{2} + p^{2} T^{4} \)
17$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 56 T^{2} + p^{2} T^{4} \)
31$C_2^2$ \( 1 + 54 T^{2} + p^{2} T^{4} \)
37$C_2^2$ \( 1 + 56 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + p T^{2} )^{2} \)
47$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 56 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
61$C_2^2$ \( 1 + 120 T^{2} + p^{2} T^{4} \)
67$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
71$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
73$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \)
79$C_2^2$ \( 1 + 86 T^{2} + p^{2} T^{4} \)
83$C_2$ \( ( 1 + 16 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 16 T + p T^{2} )^{2} \)
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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

−7.85051135596474282411277184443, −7.52048930390533064614032113918, −7.28164904888803050139720627556, −6.95863552327635429807377373134, −6.52555476265611447481759362629, −6.14958707399010275474162080463, −5.72359612518656965398308090691, −5.67147594439776898682701723090, −5.00095871547164314354255706060, −4.90135339268055628654269880337, −4.30467850308430067291832158505, −4.19616079737161093746243679368, −3.44068782709799423154077349034, −3.36271488682540483112672085126, −2.81381040486688729354367446694, −2.54151059164674333473135075065, −1.78058121932092624245849088188, −1.63377422602339157345202227179, −0.954190281293828438460045237650, −0.36513715477833542505483706902, 0.36513715477833542505483706902, 0.954190281293828438460045237650, 1.63377422602339157345202227179, 1.78058121932092624245849088188, 2.54151059164674333473135075065, 2.81381040486688729354367446694, 3.36271488682540483112672085126, 3.44068782709799423154077349034, 4.19616079737161093746243679368, 4.30467850308430067291832158505, 4.90135339268055628654269880337, 5.00095871547164314354255706060, 5.67147594439776898682701723090, 5.72359612518656965398308090691, 6.14958707399010275474162080463, 6.52555476265611447481759362629, 6.95863552327635429807377373134, 7.28164904888803050139720627556, 7.52048930390533064614032113918, 7.85051135596474282411277184443

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