Properties

Degree $4$
Conductor $5531904$
Sign $1$
Motivic weight $1$
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s − 4·5-s + 3·9-s + 4·11-s − 8·13-s − 8·15-s − 4·17-s + 4·23-s + 4·25-s + 4·27-s − 8·29-s − 8·31-s + 8·33-s − 8·37-s − 16·39-s − 4·41-s − 12·45-s − 8·51-s − 4·53-s − 16·55-s − 8·59-s − 16·61-s + 32·65-s + 8·69-s + 4·71-s − 8·73-s + 8·75-s + ⋯
L(s)  = 1  + 1.15·3-s − 1.78·5-s + 9-s + 1.20·11-s − 2.21·13-s − 2.06·15-s − 0.970·17-s + 0.834·23-s + 4/5·25-s + 0.769·27-s − 1.48·29-s − 1.43·31-s + 1.39·33-s − 1.31·37-s − 2.56·39-s − 0.624·41-s − 1.78·45-s − 1.12·51-s − 0.549·53-s − 2.15·55-s − 1.04·59-s − 2.04·61-s + 3.96·65-s + 0.963·69-s + 0.474·71-s − 0.936·73-s + 0.923·75-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5531904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5531904 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(5531904\)    =    \(2^{8} \cdot 3^{2} \cdot 7^{4}\)
Sign: $1$
Motivic weight: \(1\)
Character: induced by $\chi_{2352} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 5531904,\ (\ :1/2, 1/2),\ 1)\)

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$C_1$ \( ( 1 - T )^{2} \)
7 \( 1 \)
good5$D_{4}$ \( 1 + 4 T + 12 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
11$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
13$D_{4}$ \( 1 + 8 T + 40 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 + 4 T + 20 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \)
23$D_{4}$ \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 + 8 T + 66 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 + 8 T + 70 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
37$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
41$D_{4}$ \( 1 + 4 T + 68 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
43$C_2^2$ \( 1 + 54 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 + 86 T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
59$D_{4}$ \( 1 + 8 T + 126 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 16 T + 168 T^{2} + 16 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 102 T^{2} + p^{2} T^{4} \)
71$D_{4}$ \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 8 T + 64 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 16 T + 190 T^{2} + 16 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 8 T + 54 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 - 20 T + 260 T^{2} - 20 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 8 T + 208 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
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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

−8.698391471901117767714071677457, −8.647400233703546582468294714320, −7.72549381241748490104788887242, −7.68949541017584469638958276136, −7.30565209318904281267044274378, −7.27988486489686917125054497255, −6.54638674299259428098893164572, −6.39427980534087539741060696676, −5.36505884673792900762145014754, −5.20620313258172193691678927143, −4.43543810155034773105262990942, −4.41618109764741560377715890774, −3.76672017027205741965560454139, −3.60934246360442805807245406123, −2.98712161165684488099306228590, −2.61250149408884447892644538493, −1.80181763836107094334648188982, −1.55507447212516279169853758229, 0, 0, 1.55507447212516279169853758229, 1.80181763836107094334648188982, 2.61250149408884447892644538493, 2.98712161165684488099306228590, 3.60934246360442805807245406123, 3.76672017027205741965560454139, 4.41618109764741560377715890774, 4.43543810155034773105262990942, 5.20620313258172193691678927143, 5.36505884673792900762145014754, 6.39427980534087539741060696676, 6.54638674299259428098893164572, 7.27988486489686917125054497255, 7.30565209318904281267044274378, 7.68949541017584469638958276136, 7.72549381241748490104788887242, 8.647400233703546582468294714320, 8.698391471901117767714071677457

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