Properties

Label 4-350e2-1.1-c3e2-0-20
Degree $4$
Conductor $122500$
Sign $1$
Analytic cond. $426.450$
Root an. cond. $4.54430$
Motivic weight $3$
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  − 4·4-s − 10·9-s − 14·11-s + 16·16-s − 284·19-s − 2·29-s + 12·31-s + 40·36-s − 888·41-s + 56·44-s − 49·49-s + 324·59-s − 1.64e3·61-s − 64·64-s + 122·71-s + 1.13e3·76-s + 1.61e3·79-s − 629·81-s − 740·89-s + 140·99-s + 840·101-s − 3.32e3·109-s + 8·116-s − 2.51e3·121-s − 48·124-s + 127-s + 131-s + ⋯
L(s)  = 1  − 1/2·4-s − 0.370·9-s − 0.383·11-s + 1/4·16-s − 3.42·19-s − 0.0128·29-s + 0.0695·31-s + 5/27·36-s − 3.38·41-s + 0.191·44-s − 1/7·49-s + 0.714·59-s − 3.44·61-s − 1/8·64-s + 0.203·71-s + 1.71·76-s + 2.30·79-s − 0.862·81-s − 0.881·89-s + 0.142·99-s + 0.827·101-s − 2.91·109-s + 0.00640·116-s − 1.88·121-s − 0.0347·124-s + 0.000698·127-s + 0.000666·131-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 122500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 122500 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(122500\)    =    \(2^{2} \cdot 5^{4} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(426.450\)
Root analytic conductor: \(4.54430\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 122500,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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$C_2$ \( 1 + p^{2} T^{2} \)
5 \( 1 \)
7$C_2$ \( 1 + p^{2} T^{2} \)
good3$C_2^2$ \( 1 + 10 T^{2} + p^{6} T^{4} \)
11$C_2$ \( ( 1 + 7 T + p^{3} T^{2} )^{2} \)
13$C_2^2$ \( 1 - 22 p^{2} T^{2} + p^{6} T^{4} \)
17$C_2^2$ \( 1 - 7890 T^{2} + p^{6} T^{4} \)
19$C_2$ \( ( 1 + 142 T + p^{3} T^{2} )^{2} \)
23$C_2^2$ \( 1 - 21 p^{2} T^{2} + p^{6} T^{4} \)
29$C_2$ \( ( 1 + T + p^{3} T^{2} )^{2} \)
31$C_2$ \( ( 1 - 6 T + p^{3} T^{2} )^{2} \)
37$C_2^2$ \( 1 + 67615 T^{2} + p^{6} T^{4} \)
41$C_2$ \( ( 1 + 444 T + p^{3} T^{2} )^{2} \)
43$C_2^2$ \( 1 - 110173 T^{2} + p^{6} T^{4} \)
47$C_2^2$ \( 1 - 141082 T^{2} + p^{6} T^{4} \)
53$C_2^2$ \( 1 + 94122 T^{2} + p^{6} T^{4} \)
59$C_2$ \( ( 1 - 162 T + p^{3} T^{2} )^{2} \)
61$C_2$ \( ( 1 + 820 T + p^{3} T^{2} )^{2} \)
67$C_2^2$ \( 1 - 332165 T^{2} + p^{6} T^{4} \)
71$C_2$ \( ( 1 - 61 T + p^{3} T^{2} )^{2} \)
73$C_2^2$ \( 1 + 567566 T^{2} + p^{6} T^{4} \)
79$C_2$ \( ( 1 - 809 T + p^{3} T^{2} )^{2} \)
83$C_2^2$ \( 1 - 683890 T^{2} + p^{6} T^{4} \)
89$C_2$ \( ( 1 + 370 T + p^{3} T^{2} )^{2} \)
97$C_2^2$ \( 1 - 1729246 T^{2} + p^{6} T^{4} \)
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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

−10.57275004257815582338083879813, −10.54806900748474145379313049200, −10.07083755919642286618229564906, −9.270378927850822660539097504765, −8.985128737660146541046732562653, −8.366339114079785060689606061426, −8.248062871130949569105231580609, −7.68163388872718367174389174118, −6.67013340810291357694947453509, −6.64562447256554135563762574059, −6.01775390413292606830226858712, −5.30845656060942946316943189613, −4.80538252531654388302988518928, −4.28444484031159941744484455194, −3.71611439875552376108123221330, −2.96266288506506285229641335009, −2.19213481020705275698593034554, −1.55277467184263849010904192773, 0, 0, 1.55277467184263849010904192773, 2.19213481020705275698593034554, 2.96266288506506285229641335009, 3.71611439875552376108123221330, 4.28444484031159941744484455194, 4.80538252531654388302988518928, 5.30845656060942946316943189613, 6.01775390413292606830226858712, 6.64562447256554135563762574059, 6.67013340810291357694947453509, 7.68163388872718367174389174118, 8.248062871130949569105231580609, 8.366339114079785060689606061426, 8.985128737660146541046732562653, 9.270378927850822660539097504765, 10.07083755919642286618229564906, 10.54806900748474145379313049200, 10.57275004257815582338083879813

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