Properties

Label 4-39e4-1.1-c1e2-0-10
Degree $4$
Conductor $2313441$
Sign $1$
Analytic cond. $147.507$
Root an. cond. $3.48500$
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  − 4-s − 4·7-s − 3·16-s − 4·19-s − 10·25-s + 4·28-s − 4·31-s − 4·37-s + 16·43-s − 2·49-s − 20·61-s + 7·64-s − 28·67-s + 20·73-s + 4·76-s − 8·79-s + 20·97-s + 10·100-s − 8·103-s + 20·109-s + 12·112-s − 10·121-s + 4·124-s + 127-s + 131-s + 16·133-s + 137-s + ⋯
L(s)  = 1  − 1/2·4-s − 1.51·7-s − 3/4·16-s − 0.917·19-s − 2·25-s + 0.755·28-s − 0.718·31-s − 0.657·37-s + 2.43·43-s − 2/7·49-s − 2.56·61-s + 7/8·64-s − 3.42·67-s + 2.34·73-s + 0.458·76-s − 0.900·79-s + 2.03·97-s + 100-s − 0.788·103-s + 1.91·109-s + 1.13·112-s − 0.909·121-s + 0.359·124-s + 0.0887·127-s + 0.0873·131-s + 1.38·133-s + 0.0854·137-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2313441 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2313441 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2313441\)    =    \(3^{4} \cdot 13^{4}\)
Sign: $1$
Analytic conductor: \(147.507\)
Root analytic conductor: \(3.48500\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1521} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 2313441,\ (\ :1/2, 1/2),\ 1)\)

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)$
bad3 \( 1 \)
13 \( 1 \)
good2$C_2^2$ \( 1 + T^{2} + p^{2} T^{4} \)
5$C_2$ \( ( 1 + p T^{2} )^{2} \)
7$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
11$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
31$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 + 34 T^{2} + p^{2} T^{4} \)
43$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 + p T^{2} )^{2} \)
59$C_2^2$ \( 1 + 106 T^{2} + p^{2} T^{4} \)
61$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \)
71$C_2^2$ \( 1 + 130 T^{2} + p^{2} T^{4} \)
73$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 + 58 T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 + 130 T^{2} + p^{2} T^{4} \)
97$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
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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

−9.258486323625683228056617943448, −8.939932819757655947582294898696, −8.717394945817194771911214024179, −8.019660782712146266296032753363, −7.50122640406391094299576029293, −7.45630856642643011483008396339, −6.79963178349652217329348536959, −6.19228113557220520313158332858, −6.09472435206819771078002086054, −5.83878409238956337248852307176, −4.90789362548689044448545902106, −4.74362666343720561565526139526, −3.96158984561161604411309874545, −3.84662510345489335567849061323, −3.25564618523420746855218371578, −2.61946421446322524837121717635, −2.15683119024263477820965343635, −1.38935355750951163595138786834, 0, 0, 1.38935355750951163595138786834, 2.15683119024263477820965343635, 2.61946421446322524837121717635, 3.25564618523420746855218371578, 3.84662510345489335567849061323, 3.96158984561161604411309874545, 4.74362666343720561565526139526, 4.90789362548689044448545902106, 5.83878409238956337248852307176, 6.09472435206819771078002086054, 6.19228113557220520313158332858, 6.79963178349652217329348536959, 7.45630856642643011483008396339, 7.50122640406391094299576029293, 8.019660782712146266296032753363, 8.717394945817194771911214024179, 8.939932819757655947582294898696, 9.258486323625683228056617943448

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