Properties

Label 4-2475e2-1.1-c1e2-0-23
Degree $4$
Conductor $6125625$
Sign $1$
Analytic cond. $390.575$
Root an. cond. $4.44555$
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  − 2·2-s + 4-s + 4·7-s + 2·11-s − 8·14-s + 16-s − 8·17-s − 8·19-s − 4·22-s − 8·23-s + 4·28-s + 4·29-s + 2·32-s + 16·34-s − 12·37-s + 16·38-s − 4·41-s + 12·43-s + 2·44-s + 16·46-s − 8·47-s + 6·49-s − 4·53-s − 8·58-s + 8·59-s − 12·61-s − 11·64-s + ⋯
L(s)  = 1  − 1.41·2-s + 1/2·4-s + 1.51·7-s + 0.603·11-s − 2.13·14-s + 1/4·16-s − 1.94·17-s − 1.83·19-s − 0.852·22-s − 1.66·23-s + 0.755·28-s + 0.742·29-s + 0.353·32-s + 2.74·34-s − 1.97·37-s + 2.59·38-s − 0.624·41-s + 1.82·43-s + 0.301·44-s + 2.35·46-s − 1.16·47-s + 6/7·49-s − 0.549·53-s − 1.05·58-s + 1.04·59-s − 1.53·61-s − 1.37·64-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6125625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6125625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(6125625\)    =    \(3^{4} \cdot 5^{4} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(390.575\)
Root analytic conductor: \(4.44555\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{2475} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 6125625,\ (\ :1/2, 1/2),\ 1)\)

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 \( 1 \)
5 \( 1 \)
11$C_1$ \( ( 1 - T )^{2} \)
good2$D_{4}$ \( 1 + p T + 3 T^{2} + p^{2} T^{3} + p^{2} T^{4} \)
7$C_4$ \( 1 - 4 T + 10 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
13$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \)
17$D_{4}$ \( 1 + 8 T + 42 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 + 8 T + 46 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
23$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
29$D_{4}$ \( 1 - 4 T + 54 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
31$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$D_{4}$ \( 1 + 12 T + 78 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 + 4 T + 78 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - 12 T + 114 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
47$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
53$D_{4}$ \( 1 + 4 T - 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
59$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
61$D_{4}$ \( 1 + 12 T + 126 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 102 T^{2} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + 16 T + 174 T^{2} + 16 p T^{3} + p^{2} T^{4} \)
73$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \)
79$C_2^2$ \( 1 + 86 T^{2} + p^{2} T^{4} \)
83$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
89$D_{4}$ \( 1 - 4 T + 150 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 12 T + 198 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
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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

−8.586821501530456464839783061432, −8.513919456537368504398299507875, −8.193105262987933401903597802500, −7.916052298769149406857064929715, −7.14448861378128696754356355404, −7.09710671797099572276068316357, −6.48826185665042504057062220566, −6.04352277637619096256287229548, −5.90661617028775874847721765252, −5.07494275294566641492761933675, −4.61189264607412807046133003673, −4.41518627320243611006829144466, −4.07211395306853007804500071477, −3.44384537625838116142222658420, −2.57677189011361346570474515760, −2.20812933505978642288681380919, −1.64464510029904934324792656499, −1.32696562717415949082235892683, 0, 0, 1.32696562717415949082235892683, 1.64464510029904934324792656499, 2.20812933505978642288681380919, 2.57677189011361346570474515760, 3.44384537625838116142222658420, 4.07211395306853007804500071477, 4.41518627320243611006829144466, 4.61189264607412807046133003673, 5.07494275294566641492761933675, 5.90661617028775874847721765252, 6.04352277637619096256287229548, 6.48826185665042504057062220566, 7.09710671797099572276068316357, 7.14448861378128696754356355404, 7.916052298769149406857064929715, 8.193105262987933401903597802500, 8.513919456537368504398299507875, 8.586821501530456464839783061432

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