Properties

Degree $4$
Conductor $12446784$
Sign $1$
Motivic weight $1$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·5-s + 4·11-s − 12·17-s + 8·19-s + 4·23-s + 4·25-s + 8·37-s − 12·41-s − 8·47-s + 4·53-s − 16·55-s + 8·61-s + 4·71-s + 24·73-s + 16·79-s + 8·83-s + 48·85-s − 20·89-s − 32·95-s + 8·97-s − 4·101-s − 20·107-s + 16·109-s + 20·113-s − 16·115-s − 2·121-s + 12·125-s + ⋯
L(s)  = 1  − 1.78·5-s + 1.20·11-s − 2.91·17-s + 1.83·19-s + 0.834·23-s + 4/5·25-s + 1.31·37-s − 1.87·41-s − 1.16·47-s + 0.549·53-s − 2.15·55-s + 1.02·61-s + 0.474·71-s + 2.80·73-s + 1.80·79-s + 0.878·83-s + 5.20·85-s − 2.11·89-s − 3.28·95-s + 0.812·97-s − 0.398·101-s − 1.93·107-s + 1.53·109-s + 1.88·113-s − 1.49·115-s − 0.181·121-s + 1.07·125-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 12446784 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12446784 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(12446784\)    =    \(2^{6} \cdot 3^{4} \cdot 7^{4}\)
Sign: $1$
Motivic weight: \(1\)
Character: induced by $\chi_{3528} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 12446784,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.581781258\)
\(L(\frac12)\) \(\approx\) \(1.581781258\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 \)
good5$D_{4}$ \( 1 + 4 T + 12 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
13$C_2^2$ \( 1 + 8 T^{2} + p^{2} T^{4} \)
17$D_{4}$ \( 1 + 12 T + 4 p T^{2} + 12 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 - 8 T + 46 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
23$C_4$ \( 1 - 4 T + 42 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \)
31$C_2^2$ \( 1 + 54 T^{2} + p^{2} T^{4} \)
37$D_{4}$ \( 1 - 8 T + 58 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 + 12 T + 100 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
43$C_2^2$ \( 1 - 42 T^{2} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + 8 T + 38 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 + 46 T^{2} + p^{2} T^{4} \)
61$D_{4}$ \( 1 - 8 T + 88 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \)
71$D_{4}$ \( 1 - 4 T + 74 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 24 T + 272 T^{2} - 24 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - 16 T + 190 T^{2} - 16 p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
89$D_{4}$ \( 1 + 20 T + 260 T^{2} + 20 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 8 T + 192 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
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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

−8.548143796199831074856908565372, −8.470487959165111408562718810371, −7.966983417321824549252748107817, −7.79155479406145084558273628134, −7.09033417699355115744339926780, −6.95560104789095522114060498540, −6.72591323118778040751392443380, −6.35087459453745003815597956028, −5.77723291667095030944912990102, −5.15192048387095685455249383372, −4.90766206508253736732132804617, −4.44975168206978210696483002285, −4.06377266805380963776654642090, −3.83143138066758110584167545180, −3.18530589966462379506205237527, −3.12246299652928202966069464599, −2.10371546179764791310574554413, −1.91277585469636281637004757180, −0.875892337622378664097345436741, −0.49798518133580889681471747888, 0.49798518133580889681471747888, 0.875892337622378664097345436741, 1.91277585469636281637004757180, 2.10371546179764791310574554413, 3.12246299652928202966069464599, 3.18530589966462379506205237527, 3.83143138066758110584167545180, 4.06377266805380963776654642090, 4.44975168206978210696483002285, 4.90766206508253736732132804617, 5.15192048387095685455249383372, 5.77723291667095030944912990102, 6.35087459453745003815597956028, 6.72591323118778040751392443380, 6.95560104789095522114060498540, 7.09033417699355115744339926780, 7.79155479406145084558273628134, 7.966983417321824549252748107817, 8.470487959165111408562718810371, 8.548143796199831074856908565372

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