Properties

Label 4-96e4-1.1-c1e2-0-18
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

−7.36612501016609359836750826425, −7.34360028865030177147759632854, −6.60202261268425446887867195214, −6.45206427923396380051247402916, −6.30640441217940420117602190026, −6.16560180748490936870027295396, −5.46852623312277093154441666002, −5.10095702426398963410622284536, −4.62822578227172181092042021385, −4.43619876001694472641317193489, −3.93732789454797866047597312211, −3.77262608587537336018732660326, −3.23197142822813961774232483768, −3.07926334385005403221238299790, −2.13840734500914972626742258814, −1.94811419374731618089291410793, −1.49487605456622995793581833708, −1.21060100498829016254849936091, 0, 0, 1.21060100498829016254849936091, 1.49487605456622995793581833708, 1.94811419374731618089291410793, 2.13840734500914972626742258814, 3.07926334385005403221238299790, 3.23197142822813961774232483768, 3.77262608587537336018732660326, 3.93732789454797866047597312211, 4.43619876001694472641317193489, 4.62822578227172181092042021385, 5.10095702426398963410622284536, 5.46852623312277093154441666002, 6.16560180748490936870027295396, 6.30640441217940420117602190026, 6.45206427923396380051247402916, 6.60202261268425446887867195214, 7.34360028865030177147759632854, 7.36612501016609359836750826425

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