Properties

Label 4-90e2-1.1-c2e2-0-2
Degree $4$
Conductor $8100$
Sign $1$
Analytic cond. $6.01388$
Root an. cond. $1.56598$
Motivic weight $2$
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 + 2·4-s + 6·5-s + 16·7-s + 12·10-s − 8·11-s − 6·13-s + 32·14-s − 4·16-s − 38·17-s + 12·20-s − 16·22-s + 40·23-s + 11·25-s − 12·26-s + 32·28-s − 88·31-s − 8·32-s − 76·34-s + 96·35-s − 6·37-s − 140·41-s + 72·43-s − 16·44-s + 80·46-s + 128·49-s + 22·50-s + ⋯
L(s)  = 1  + 2-s + 1/2·4-s + 6/5·5-s + 16/7·7-s + 6/5·10-s − 0.727·11-s − 0.461·13-s + 16/7·14-s − 1/4·16-s − 2.23·17-s + 3/5·20-s − 0.727·22-s + 1.73·23-s + 0.439·25-s − 0.461·26-s + 8/7·28-s − 2.83·31-s − 1/4·32-s − 2.23·34-s + 2.74·35-s − 0.162·37-s − 3.41·41-s + 1.67·43-s − 0.363·44-s + 1.73·46-s + 2.61·49-s + 0.439·50-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8100 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(8100\)    =    \(2^{2} \cdot 3^{4} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(6.01388\)
Root analytic conductor: \(1.56598\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 8100,\ (\ :1, 1),\ 1)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(3.395991236\)
\(L(\frac12)\) \(\approx\) \(3.395991236\)
\(L(2)\) 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$C_2$ \( 1 - p T + p T^{2} \)
3 \( 1 \)
5$C_2$ \( 1 - 6 T + p^{2} T^{2} \)
good7$C_2^2$ \( 1 - 16 T + 128 T^{2} - 16 p^{2} T^{3} + p^{4} T^{4} \)
11$C_2$ \( ( 1 + 4 T + p^{2} T^{2} )^{2} \)
13$C_2^2$ \( 1 + 6 T + 18 T^{2} + 6 p^{2} T^{3} + p^{4} T^{4} \)
17$C_2^2$ \( 1 + 38 T + 722 T^{2} + 38 p^{2} T^{3} + p^{4} T^{4} \)
19$C_2^2$ \( 1 - 658 T^{2} + p^{4} T^{4} \)
23$C_2^2$ \( 1 - 40 T + 800 T^{2} - 40 p^{2} T^{3} + p^{4} T^{4} \)
29$C_2^2$ \( 1 - 238 T^{2} + p^{4} T^{4} \)
31$C_2$ \( ( 1 + 44 T + p^{2} T^{2} )^{2} \)
37$C_2^2$ \( 1 + 6 T + 18 T^{2} + 6 p^{2} T^{3} + p^{4} T^{4} \)
41$C_2$ \( ( 1 + 70 T + p^{2} T^{2} )^{2} \)
43$C_2^2$ \( 1 - 72 T + 2592 T^{2} - 72 p^{2} T^{3} + p^{4} T^{4} \)
47$C_2^2$ \( 1 + p^{4} T^{4} \)
53$C_2$ \( ( 1 - 56 T + p^{2} T^{2} )( 1 + 90 T + p^{2} T^{2} ) \)
59$C_2^2$ \( 1 + 1502 T^{2} + p^{4} T^{4} \)
61$C_2$ \( ( 1 - 72 T + p^{2} T^{2} )^{2} \)
67$C_2^2$ \( 1 - 88 T + 3872 T^{2} - 88 p^{2} T^{3} + p^{4} T^{4} \)
71$C_2$ \( ( 1 - 88 T + p^{2} T^{2} )^{2} \)
73$C_2^2$ \( 1 - 110 T + 6050 T^{2} - 110 p^{2} T^{3} + p^{4} T^{4} \)
79$C_2^2$ \( 1 - 12338 T^{2} + p^{4} T^{4} \)
83$C_2^2$ \( 1 - 48 T + 1152 T^{2} - 48 p^{2} T^{3} + p^{4} T^{4} \)
89$C_2^2$ \( 1 - 15166 T^{2} + p^{4} T^{4} \)
97$C_2^2$ \( 1 + 114 T + 6498 T^{2} + 114 p^{2} T^{3} + p^{4} T^{4} \)
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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

−14.04069499655929341045975085706, −13.71462172595389698386510033502, −12.99268275993724894539496173295, −12.94633890994636546402095128105, −12.08521945752718196173176159383, −11.23881236266706286156187677290, −11.00036747124215757976324621525, −10.81907659011399430574968761461, −9.752002988786092542843397553355, −9.112987347504254366248898166486, −8.569935515762549845422525506963, −7.986747517728860552229865650803, −7.00050538449587226768249018330, −6.73231911598271029206748655992, −5.38090472814699909208613571862, −5.23742088478129338331876892094, −4.81756719324260560081254253982, −3.79411676590324652133929518839, −2.29470844174315848453011874901, −1.90007058324373184107056649676, 1.90007058324373184107056649676, 2.29470844174315848453011874901, 3.79411676590324652133929518839, 4.81756719324260560081254253982, 5.23742088478129338331876892094, 5.38090472814699909208613571862, 6.73231911598271029206748655992, 7.00050538449587226768249018330, 7.986747517728860552229865650803, 8.569935515762549845422525506963, 9.112987347504254366248898166486, 9.752002988786092542843397553355, 10.81907659011399430574968761461, 11.00036747124215757976324621525, 11.23881236266706286156187677290, 12.08521945752718196173176159383, 12.94633890994636546402095128105, 12.99268275993724894539496173295, 13.71462172595389698386510033502, 14.04069499655929341045975085706

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