Properties

Label 4-4608e2-1.1-c1e2-0-34
Degree $4$
Conductor $21233664$
Sign $1$
Analytic cond. $1353.87$
Root an. cond. $6.06589$
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  − 4·13-s + 12·23-s + 2·25-s + 4·37-s + 20·47-s + 12·49-s + 24·59-s − 20·61-s + 4·71-s − 20·73-s + 8·83-s + 24·97-s − 16·107-s + 20·109-s − 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 14·169-s + 173-s + 179-s + ⋯
L(s)  = 1  − 1.10·13-s + 2.50·23-s + 2/5·25-s + 0.657·37-s + 2.91·47-s + 12/7·49-s + 3.12·59-s − 2.56·61-s + 0.474·71-s − 2.34·73-s + 0.878·83-s + 2.43·97-s − 1.54·107-s + 1.91·109-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 − 1.07·169-s + 0.0760·173-s + 0.0747·179-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 21233664 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21233664 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(21233664\)    =    \(2^{18} \cdot 3^{4}\)
Sign: $1$
Analytic conductor: \(1353.87\)
Root analytic conductor: \(6.06589\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{4608} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 21233664,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.257370792\)
\(L(\frac12)\) \(\approx\) \(3.257370792\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
7$C_2^2$ \( 1 - 12 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
17$C_2^2$ \( 1 - 32 T^{2} + p^{2} T^{4} \)
19$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \)
23$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
29$C_2^2$ \( 1 - 26 T^{2} + p^{2} T^{4} \)
31$C_2^2$ \( 1 + 36 T^{2} + p^{2} T^{4} \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 - 64 T^{2} + p^{2} T^{4} \)
43$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
47$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
53$C_2^2$ \( 1 - 74 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 102 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
79$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \)
83$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 - 80 T^{2} + p^{2} T^{4} \)
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

−8.541348756136689073782896055379, −8.243407613607398816545171528552, −7.47226591349202500757507217901, −7.40308398296919935030251549440, −7.16182202388232032733600562585, −6.91035365787776931389898515052, −6.28344220831121769811173796880, −5.82619666635213650403729242706, −5.68554834163475471066921613941, −5.07564062504294069323065310653, −4.79630525312801323286761518690, −4.55579746025051230656506941528, −3.90067216209622636299953080181, −3.67320386780717945841491230456, −2.89387165350341469181261838315, −2.71322142207489615024343636441, −2.36503840258358066388202104257, −1.67080124538445968311198073427, −0.892789921136727431852781688924, −0.64711876259644467384693553227, 0.64711876259644467384693553227, 0.892789921136727431852781688924, 1.67080124538445968311198073427, 2.36503840258358066388202104257, 2.71322142207489615024343636441, 2.89387165350341469181261838315, 3.67320386780717945841491230456, 3.90067216209622636299953080181, 4.55579746025051230656506941528, 4.79630525312801323286761518690, 5.07564062504294069323065310653, 5.68554834163475471066921613941, 5.82619666635213650403729242706, 6.28344220831121769811173796880, 6.91035365787776931389898515052, 7.16182202388232032733600562585, 7.40308398296919935030251549440, 7.47226591349202500757507217901, 8.243407613607398816545171528552, 8.541348756136689073782896055379

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