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  − 2·5-s − 6·11-s + 6·13-s − 4·17-s − 5·19-s − 4·23-s + 5·25-s + 8·29-s + 7·31-s + 9·37-s − 4·41-s − 2·43-s − 2·47-s + 8·53-s + 12·55-s + 10·61-s − 12·65-s + 15·67-s + 12·71-s − 11·73-s − 79-s + 12·83-s + 8·85-s + 8·89-s + 10·95-s + 28·97-s + 6·101-s + ⋯
L(s)  = 1  − 0.894·5-s − 1.80·11-s + 1.66·13-s − 0.970·17-s − 1.14·19-s − 0.834·23-s + 25-s + 1.48·29-s + 1.25·31-s + 1.47·37-s − 0.624·41-s − 0.304·43-s − 0.291·47-s + 1.09·53-s + 1.61·55-s + 1.28·61-s − 1.48·65-s + 1.83·67-s + 1.42·71-s − 1.28·73-s − 0.112·79-s + 1.31·83-s + 0.867·85-s + 0.847·89-s + 1.02·95-s + 2.84·97-s + 0.597·101-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

\(L(1)\) \(\approx\) \(1.859240734\)
\(L(\frac12)\) \(\approx\) \(1.859240734\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 + 2 T - T^{2} + 2 p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + 6 T + 25 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
17$C_2^2$ \( 1 + 4 T - T^{2} + 4 p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 + 5 T + 6 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 + 4 T - 7 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
37$C_2^2$ \( 1 - 9 T + 44 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 + 2 T - 43 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 8 T + 11 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 10 T + 39 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 - 15 T + 158 T^{2} - 15 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 + 11 T + 48 T^{2} + 11 p T^{3} + p^{2} T^{4} \)
79$C_2^2$ \( 1 + T - 78 T^{2} + p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 - 8 T - 25 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
97$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \)
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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

−8.513785836516523715034270415574, −8.177142072212803551387218772635, −8.133500421065872882683525161474, −8.014107016678255440021689612767, −7.25071342869270702159529916655, −6.87023499626272756386024687317, −6.39171474401558042544198356079, −6.37081219292544425564948666557, −5.82086716386566217178480872908, −5.25422803076883456198732105243, −4.93758288831640695408973174555, −4.48781435053853499175980300544, −4.15624859052824832937202198813, −3.76292435624383440991346307926, −3.25644039027928428083143820484, −2.73014992314598800075977077435, −2.36524110087633703527091492323, −1.89682433560961563589992382436, −0.864726339517738493634694224314, −0.53734962439645679382839310971, 0.53734962439645679382839310971, 0.864726339517738493634694224314, 1.89682433560961563589992382436, 2.36524110087633703527091492323, 2.73014992314598800075977077435, 3.25644039027928428083143820484, 3.76292435624383440991346307926, 4.15624859052824832937202198813, 4.48781435053853499175980300544, 4.93758288831640695408973174555, 5.25422803076883456198732105243, 5.82086716386566217178480872908, 6.37081219292544425564948666557, 6.39171474401558042544198356079, 6.87023499626272756386024687317, 7.25071342869270702159529916655, 8.014107016678255440021689612767, 8.133500421065872882683525161474, 8.177142072212803551387218772635, 8.513785836516523715034270415574

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