Properties

Label 4-4840e2-1.1-c1e2-0-5
Degree $4$
Conductor $23425600$
Sign $1$
Analytic cond. $1493.63$
Root an. cond. $6.21671$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 2·5-s + 7-s − 9-s − 4·13-s − 2·15-s − 3·17-s − 19-s − 21-s + 2·23-s + 3·25-s − 7·29-s + 9·31-s + 2·35-s + 37-s + 4·39-s − 4·41-s − 4·43-s − 2·45-s − 18·47-s − 9·49-s + 3·51-s + 5·53-s + 57-s − 6·59-s − 17·61-s − 63-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.894·5-s + 0.377·7-s − 1/3·9-s − 1.10·13-s − 0.516·15-s − 0.727·17-s − 0.229·19-s − 0.218·21-s + 0.417·23-s + 3/5·25-s − 1.29·29-s + 1.61·31-s + 0.338·35-s + 0.164·37-s + 0.640·39-s − 0.624·41-s − 0.609·43-s − 0.298·45-s − 2.62·47-s − 9/7·49-s + 0.420·51-s + 0.686·53-s + 0.132·57-s − 0.781·59-s − 2.17·61-s − 0.125·63-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 23425600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23425600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(23425600\)    =    \(2^{6} \cdot 5^{2} \cdot 11^{4}\)
Sign: $1$
Analytic conductor: \(1493.63\)
Root analytic conductor: \(6.21671\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 23425600,\ (\ :1/2, 1/2),\ 1)\)

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 \( 1 \)
5$C_1$ \( ( 1 - T )^{2} \)
11 \( 1 \)
good3$D_{4}$ \( 1 + T + 2 T^{2} + p T^{3} + p^{2} T^{4} \)
7$D_{4}$ \( 1 - T + 10 T^{2} - p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
17$D_{4}$ \( 1 + 3 T + 32 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 + T + 34 T^{2} + p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 - 2 T + 30 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 + 7 T + 32 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 - 9 T + 78 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 - T + 36 T^{2} - p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
43$D_{4}$ \( 1 + 4 T + 22 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + 18 T + 158 T^{2} + 18 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 - 5 T + 108 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 6 T + 110 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 17 T + 156 T^{2} + 17 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + 4 T + 70 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + 5 T + 110 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
73$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
79$D_{4}$ \( 1 - 6 T + 14 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 2 T + 150 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 21 T + 284 T^{2} + 21 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 6 T + 186 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
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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

−7.933776508072419332205590843962, −7.86469244371202825964927119621, −7.34791980990579251693186990643, −6.89750292623300064372994407118, −6.51520881569655917677472967705, −6.36882267036271177289066026469, −5.92878150060719532719106845289, −5.53566517976738044056699672466, −5.07957495107868361671641249561, −4.84970650673894248386307796477, −4.59984757797106266493448470323, −4.10666770116848453513515832747, −3.41133884661565680844818856457, −3.02269310613323189895054397502, −2.63767762401214439566065503510, −2.10885078949032875362994811415, −1.60978107931205261013769993541, −1.27608616511820646456767353166, 0, 0, 1.27608616511820646456767353166, 1.60978107931205261013769993541, 2.10885078949032875362994811415, 2.63767762401214439566065503510, 3.02269310613323189895054397502, 3.41133884661565680844818856457, 4.10666770116848453513515832747, 4.59984757797106266493448470323, 4.84970650673894248386307796477, 5.07957495107868361671641249561, 5.53566517976738044056699672466, 5.92878150060719532719106845289, 6.36882267036271177289066026469, 6.51520881569655917677472967705, 6.89750292623300064372994407118, 7.34791980990579251693186990643, 7.86469244371202825964927119621, 7.933776508072419332205590843962

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