Properties

Label 4-2352e2-1.1-c1e2-0-53
Degree $4$
Conductor $5531904$
Sign $1$
Analytic cond. $352.718$
Root an. cond. $4.33368$
Motivic weight $1$
Arithmetic yes
Rational yes
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 + 8·15-s − 12·17-s + 8·19-s + 4·23-s + 4·25-s − 4·27-s − 8·33-s + 8·37-s − 12·41-s − 12·45-s + 8·47-s + 24·51-s − 4·53-s − 16·55-s − 16·57-s − 8·61-s − 8·69-s + 4·71-s − 24·73-s − 8·75-s − 16·79-s + 5·81-s − 8·83-s + ⋯
L(s)  = 1  − 1.15·3-s − 1.78·5-s + 9-s + 1.20·11-s + 2.06·15-s − 2.91·17-s + 1.83·19-s + 0.834·23-s + 4/5·25-s − 0.769·27-s − 1.39·33-s + 1.31·37-s − 1.87·41-s − 1.78·45-s + 1.16·47-s + 3.36·51-s − 0.549·53-s − 2.15·55-s − 2.11·57-s − 1.02·61-s − 0.963·69-s + 0.474·71-s − 2.80·73-s − 0.923·75-s − 1.80·79-s + 5/9·81-s − 0.878·83-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: $1$
Analytic conductor: \(352.718\)
Root analytic conductor: \(4.33368\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
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$D_{4}$ \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
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

−8.666309315043024180010044936232, −8.633896624027348567285064966310, −7.72806953602727379737167512769, −7.70170300688637950129612773536, −7.10276400475580769464366541150, −6.96984238991792902771881241245, −6.40017318611099806179872290760, −6.34517138040762474760387129009, −5.48100953375930169935458358780, −5.32470870015525549358797508304, −4.54388982970115817675053300559, −4.45508437457356945914192701442, −3.98060992217025302004451488127, −3.79029730786822254235881330119, −2.93187929445904833014945174614, −2.63617475241101087708365634860, −1.47016836472531453831984947977, −1.25671956635564698268715408649, 0, 0, 1.25671956635564698268715408649, 1.47016836472531453831984947977, 2.63617475241101087708365634860, 2.93187929445904833014945174614, 3.79029730786822254235881330119, 3.98060992217025302004451488127, 4.45508437457356945914192701442, 4.54388982970115817675053300559, 5.32470870015525549358797508304, 5.48100953375930169935458358780, 6.34517138040762474760387129009, 6.40017318611099806179872290760, 6.96984238991792902771881241245, 7.10276400475580769464366541150, 7.70170300688637950129612773536, 7.72806953602727379737167512769, 8.633896624027348567285064966310, 8.666309315043024180010044936232

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