Properties

Label 4-3675e2-1.1-c1e2-0-13
Degree $4$
Conductor $13505625$
Sign $1$
Analytic cond. $861.130$
Root an. cond. $5.41710$
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-s − 2·3-s − 2·4-s + 2·6-s + 3·8-s + 3·9-s + 4·12-s + 4·13-s + 16-s + 4·17-s − 3·18-s − 4·19-s − 4·23-s − 6·24-s − 4·26-s − 4·27-s − 6·29-s − 8·31-s − 2·32-s − 4·34-s − 6·36-s − 2·37-s + 4·38-s − 8·39-s + 4·43-s + 4·46-s + 12·47-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.15·3-s − 4-s + 0.816·6-s + 1.06·8-s + 9-s + 1.15·12-s + 1.10·13-s + 1/4·16-s + 0.970·17-s − 0.707·18-s − 0.917·19-s − 0.834·23-s − 1.22·24-s − 0.784·26-s − 0.769·27-s − 1.11·29-s − 1.43·31-s − 0.353·32-s − 0.685·34-s − 36-s − 0.328·37-s + 0.648·38-s − 1.28·39-s + 0.609·43-s + 0.589·46-s + 1.75·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 13505625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13505625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(13505625\)    =    \(3^{2} \cdot 5^{4} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(861.130\)
Root analytic conductor: \(5.41710\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 13505625,\ (\ :1/2, 1/2),\ 1)\)

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$C_1$ \( ( 1 + T )^{2} \)
5 \( 1 \)
7 \( 1 \)
good2$D_{4}$ \( 1 + T + 3 T^{2} + p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + 17 T^{2} + p^{2} T^{4} \)
13$D_{4}$ \( 1 - 4 T + 10 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 + 4 T + 22 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 + 4 T + 45 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
37$D_{4}$ \( 1 + 2 T + 55 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
41$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - 4 T + 45 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 - 12 T + 110 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
59$D_{4}$ \( 1 + 4 T + 102 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
61$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
67$D_{4}$ \( 1 - 12 T + 125 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 - 12 T + 173 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 20 T + 226 T^{2} + 20 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - 8 T + 129 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 16 T + 150 T^{2} - 16 p T^{3} + p^{2} T^{4} \)
89$C_4$ \( 1 + 12 T + 194 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
97$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
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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

−8.244112212624253705309480169298, −8.113757456482280977035150045005, −7.62212699278570071919436946292, −7.40714228633650694392930653106, −6.82155388824115498873885779416, −6.49838167717757978207073511902, −6.01067829036703858573026269824, −5.70640472968556350965219423705, −5.28535548361277769017829210721, −5.24702627027636065080606666903, −4.31208761114218240125196099571, −4.22808665355881109669952588743, −3.82416067840142963895123082571, −3.51068305168562228887630606089, −2.63877929786635850219665808762, −2.06750539763685971804813932702, −1.26854441787210996771724716775, −1.15578551851287966772301927106, 0, 0, 1.15578551851287966772301927106, 1.26854441787210996771724716775, 2.06750539763685971804813932702, 2.63877929786635850219665808762, 3.51068305168562228887630606089, 3.82416067840142963895123082571, 4.22808665355881109669952588743, 4.31208761114218240125196099571, 5.24702627027636065080606666903, 5.28535548361277769017829210721, 5.70640472968556350965219423705, 6.01067829036703858573026269824, 6.49838167717757978207073511902, 6.82155388824115498873885779416, 7.40714228633650694392930653106, 7.62212699278570071919436946292, 8.113757456482280977035150045005, 8.244112212624253705309480169298

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