Properties

Label 4-1950e2-1.1-c1e2-0-36
Degree $4$
Conductor $3802500$
Sign $1$
Analytic cond. $242.450$
Root an. cond. $3.94598$
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  + 2·2-s + 3·4-s − 4·7-s + 4·8-s − 9-s + 4·13-s − 8·14-s + 5·16-s − 2·18-s + 8·26-s − 12·28-s + 20·29-s + 6·32-s − 3·36-s + 16·37-s − 24·47-s − 2·49-s + 12·52-s − 16·56-s + 40·58-s + 4·61-s + 4·63-s + 7·64-s − 4·67-s − 4·72-s + 8·73-s + 32·74-s + ⋯
L(s)  = 1  + 1.41·2-s + 3/2·4-s − 1.51·7-s + 1.41·8-s − 1/3·9-s + 1.10·13-s − 2.13·14-s + 5/4·16-s − 0.471·18-s + 1.56·26-s − 2.26·28-s + 3.71·29-s + 1.06·32-s − 1/2·36-s + 2.63·37-s − 3.50·47-s − 2/7·49-s + 1.66·52-s − 2.13·56-s + 5.25·58-s + 0.512·61-s + 0.503·63-s + 7/8·64-s − 0.488·67-s − 0.471·72-s + 0.936·73-s + 3.71·74-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3802500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3802500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(3802500\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{4} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(242.450\)
Root analytic conductor: \(3.94598\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 3802500,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(5.553682416\)
\(L(\frac12)\) \(\approx\) \(5.553682416\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{2} \)
3$C_2$ \( 1 + T^{2} \)
5 \( 1 \)
13$C_2$ \( 1 - 4 T + p T^{2} \)
good7$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
11$C_2$ \( ( 1 - p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
19$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \)
37$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
43$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
47$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
53$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 102 T^{2} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 - 142 T^{2} + p^{2} T^{4} \)
97$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
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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

−9.694016712378035754981064154288, −8.998056782952174325958252430631, −8.366838030368023499126782718250, −8.137369780441600156200967963196, −7.997860966368313205547891046796, −7.14150648635048845759919357393, −6.63267108306935304094952216906, −6.55678940095100445499472169153, −6.16902622843767545765323769256, −6.02779769269554767986543824390, −5.29884684416494913216419414403, −4.86735091020590884769521277253, −4.45720548217776739254632501244, −4.11468861395812635048779299946, −3.39699469984014380832398708088, −3.04254088241409638182579670218, −2.96608493158294955620640159852, −2.25701906041595830660196857955, −1.39269502289205923453683145862, −0.69984946794002142280064562161, 0.69984946794002142280064562161, 1.39269502289205923453683145862, 2.25701906041595830660196857955, 2.96608493158294955620640159852, 3.04254088241409638182579670218, 3.39699469984014380832398708088, 4.11468861395812635048779299946, 4.45720548217776739254632501244, 4.86735091020590884769521277253, 5.29884684416494913216419414403, 6.02779769269554767986543824390, 6.16902622843767545765323769256, 6.55678940095100445499472169153, 6.63267108306935304094952216906, 7.14150648635048845759919357393, 7.997860966368313205547891046796, 8.137369780441600156200967963196, 8.366838030368023499126782718250, 8.998056782952174325958252430631, 9.694016712378035754981064154288

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