Properties

Label 4-48e4-1.1-c2e2-0-6
Degree $4$
Conductor $5308416$
Sign $1$
Analytic cond. $3941.25$
Root an. cond. $7.92334$
Motivic weight $2$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 8·5-s − 40·13-s + 20·17-s − 2·25-s + 40·29-s − 40·37-s − 60·41-s − 30·49-s − 120·53-s + 56·61-s − 320·65-s + 20·73-s + 160·85-s + 44·89-s + 300·97-s − 280·101-s − 136·109-s + 380·113-s + 42·121-s − 344·125-s + 127-s + 131-s + 137-s + 139-s + 320·145-s + 149-s + 151-s + ⋯
L(s)  = 1  + 8/5·5-s − 3.07·13-s + 1.17·17-s − 0.0799·25-s + 1.37·29-s − 1.08·37-s − 1.46·41-s − 0.612·49-s − 2.26·53-s + 0.918·61-s − 4.92·65-s + 0.273·73-s + 1.88·85-s + 0.494·89-s + 3.09·97-s − 2.77·101-s − 1.24·109-s + 3.36·113-s + 0.347·121-s − 2.75·125-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 2.20·145-s + 0.00671·149-s + 0.00662·151-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5308416 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5308416 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(5308416\)    =    \(2^{16} \cdot 3^{4}\)
Sign: $1$
Analytic conductor: \(3941.25\)
Root analytic conductor: \(7.92334\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 5308416,\ (\ :1, 1),\ 1)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.327952945\)
\(L(\frac12)\) \(\approx\) \(1.327952945\)
\(L(2)\) 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 \( 1 \)
3 \( 1 \)
good5$C_2$ \( ( 1 - 4 T + p^{2} T^{2} )^{2} \)
7$C_2^2$ \( 1 + 30 T^{2} + p^{4} T^{4} \)
11$C_2^2$ \( 1 - 42 T^{2} + p^{4} T^{4} \)
13$C_2$ \( ( 1 + 20 T + p^{2} T^{2} )^{2} \)
17$C_2$ \( ( 1 - 10 T + p^{2} T^{2} )^{2} \)
19$C_2^2$ \( 1 - 522 T^{2} + p^{4} T^{4} \)
23$C_2^2$ \( 1 - 930 T^{2} + p^{4} T^{4} \)
29$C_2$ \( ( 1 - 20 T + p^{2} T^{2} )^{2} \)
31$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
37$C_2$ \( ( 1 + 20 T + p^{2} T^{2} )^{2} \)
41$C_2$ \( ( 1 + 30 T + p^{2} T^{2} )^{2} \)
43$C_2^2$ \( 1 - 3690 T^{2} + p^{4} T^{4} \)
47$C_2^2$ \( 1 + 190 T^{2} + p^{4} T^{4} \)
53$C_2$ \( ( 1 + 60 T + p^{2} T^{2} )^{2} \)
59$C_2^2$ \( 1 - 5162 T^{2} + p^{4} T^{4} \)
61$C_2$ \( ( 1 - 28 T + p^{2} T^{2} )^{2} \)
67$C_2^2$ \( 1 - 2250 T^{2} + p^{4} T^{4} \)
71$C_2^2$ \( 1 - 6882 T^{2} + p^{4} T^{4} \)
73$C_2$ \( ( 1 - 10 T + p^{2} T^{2} )^{2} \)
79$C_2^2$ \( 1 + 318 T^{2} + p^{4} T^{4} \)
83$C_2^2$ \( 1 - 13130 T^{2} + p^{4} T^{4} \)
89$C_2$ \( ( 1 - 22 T + p^{2} T^{2} )^{2} \)
97$C_2$ \( ( 1 - 150 T + p^{2} T^{2} )^{2} \)
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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

−9.402855138005648185733140638634, −8.515247026924144646871012321254, −8.354295589857998198789381741501, −7.67484450851540879447917889324, −7.61970694265595076208826305486, −6.90266002064276253083775873820, −6.86999329019866910000899011201, −6.18328611624371729491969128342, −5.94771889625528951630778354642, −5.28545940972996193274178889444, −5.20236194843611335452499509372, −4.72606280787076674608529276111, −4.48715743459351750163321803923, −3.43824380353790770137270608513, −3.28362910829426500296302983912, −2.55988177325439034870744817903, −2.22297005995156782541026144105, −1.83646567609845921241492403065, −1.23055585108132590394560203468, −0.26670275087806080278407233906, 0.26670275087806080278407233906, 1.23055585108132590394560203468, 1.83646567609845921241492403065, 2.22297005995156782541026144105, 2.55988177325439034870744817903, 3.28362910829426500296302983912, 3.43824380353790770137270608513, 4.48715743459351750163321803923, 4.72606280787076674608529276111, 5.20236194843611335452499509372, 5.28545940972996193274178889444, 5.94771889625528951630778354642, 6.18328611624371729491969128342, 6.86999329019866910000899011201, 6.90266002064276253083775873820, 7.61970694265595076208826305486, 7.67484450851540879447917889324, 8.354295589857998198789381741501, 8.515247026924144646871012321254, 9.402855138005648185733140638634

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