Properties

Label 4-95e4-1.1-c1e2-0-0
Degree $4$
Conductor $81450625$
Sign $1$
Analytic cond. $5193.36$
Root an. cond. $8.48910$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·4-s − 6·9-s + 10·11-s + 12·16-s + 24·36-s − 40·44-s + 5·49-s − 30·61-s − 32·64-s + 27·81-s − 60·99-s − 20·101-s + 53·121-s + 127-s + 131-s + 137-s + 139-s − 72·144-s + 149-s + 151-s + 157-s + 163-s + 167-s − 26·169-s + 173-s + 120·176-s + 179-s + ⋯
L(s)  = 1  − 2·4-s − 2·9-s + 3.01·11-s + 3·16-s + 4·36-s − 6.03·44-s + 5/7·49-s − 3.84·61-s − 4·64-s + 3·81-s − 6.03·99-s − 1.99·101-s + 4.81·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 6·144-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 2·169-s + 0.0760·173-s + 9.04·176-s + 0.0747·179-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 81450625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81450625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(81450625\)    =    \(5^{4} \cdot 19^{4}\)
Sign: $1$
Analytic conductor: \(5193.36\)
Root analytic conductor: \(8.48910\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 81450625,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7393982934\)
\(L(\frac12)\) \(\approx\) \(0.7393982934\)
\(L(\frac{3}{2})\) 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)$
bad5 \( 1 \)
19 \( 1 \)
good2$C_2$ \( ( 1 + p T^{2} )^{2} \)
3$C_2$ \( ( 1 + p T^{2} )^{2} \)
7$C_2^2$ \( 1 - 5 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 + p T^{2} )^{2} \)
17$C_2^2$ \( 1 + 15 T^{2} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$C_2$ \( ( 1 + p T^{2} )^{2} \)
41$C_2$ \( ( 1 + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 85 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 + 75 T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 + p T^{2} )^{2} \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 15 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 25 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + p T^{2} )^{2} \)
83$C_2^2$ \( 1 + 90 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + p T^{2} )^{2} \)
97$C_2$ \( ( 1 + p T^{2} )^{2} \)
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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

−7.87228133581023497482038977982, −7.84163220166295224049137351703, −7.26219499095555077411811779446, −6.82889104615404670396933503724, −6.24537487885028307234805971327, −6.19312827325096885217432854483, −5.93752055122709237833610759312, −5.50653509629120021647298524263, −4.98090486488464700170258968115, −4.88929259409178355287874676243, −4.30400428808585718401726197552, −4.05796251768825214799445401206, −3.61122366219817124898056168815, −3.57905045340492475134955966368, −2.86826313090269554247123305551, −2.70479366693480888803891032607, −1.66489463975726893291534958192, −1.41705652399818893920402706597, −0.883885917930079300455782817475, −0.26297038550728600901815315916, 0.26297038550728600901815315916, 0.883885917930079300455782817475, 1.41705652399818893920402706597, 1.66489463975726893291534958192, 2.70479366693480888803891032607, 2.86826313090269554247123305551, 3.57905045340492475134955966368, 3.61122366219817124898056168815, 4.05796251768825214799445401206, 4.30400428808585718401726197552, 4.88929259409178355287874676243, 4.98090486488464700170258968115, 5.50653509629120021647298524263, 5.93752055122709237833610759312, 6.19312827325096885217432854483, 6.24537487885028307234805971327, 6.82889104615404670396933503724, 7.26219499095555077411811779446, 7.84163220166295224049137351703, 7.87228133581023497482038977982

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