Properties

Label 4-1792e2-1.1-c1e2-0-2
Degree $4$
Conductor $3211264$
Sign $1$
Analytic cond. $204.752$
Root an. cond. $3.78274$
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  + 2·7-s + 2·9-s − 4·17-s − 16·23-s + 10·25-s − 8·31-s + 20·41-s + 8·47-s + 3·49-s + 4·63-s + 12·73-s − 32·79-s − 5·81-s − 36·89-s − 4·97-s + 8·103-s + 12·113-s − 8·119-s + 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 8·153-s + 157-s + ⋯
L(s)  = 1  + 0.755·7-s + 2/3·9-s − 0.970·17-s − 3.33·23-s + 2·25-s − 1.43·31-s + 3.12·41-s + 1.16·47-s + 3/7·49-s + 0.503·63-s + 1.40·73-s − 3.60·79-s − 5/9·81-s − 3.81·89-s − 0.406·97-s + 0.788·103-s + 1.12·113-s − 0.733·119-s + 6/11·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s − 0.646·153-s + 0.0798·157-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3211264 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3211264 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(3211264\)    =    \(2^{16} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(204.752\)
Root analytic conductor: \(3.78274\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1792} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 3211264,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.110743531\)
\(L(\frac12)\) \(\approx\) \(2.110743531\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7$C_1$ \( ( 1 - T )^{2} \)
good3$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
5$C_2$ \( ( 1 - p T^{2} )^{2} \)
11$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
17$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
19$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
23$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
29$C_2^2$ \( 1 - 54 T^{2} + p^{2} T^{4} \)
31$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
47$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
53$C_2^2$ \( 1 - 102 T^{2} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \)
67$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 + 16 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 162 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 18 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
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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

−9.672732593597194950984025557283, −8.886822846202992888782878355008, −8.785723803933609650617371643745, −8.291998783204522234294349930134, −7.946775133717328822803951042244, −7.39154135809763517554922142448, −7.23902651652528035790225865926, −6.82433041023246349280288512703, −6.14704461923121500029274219723, −5.87072247714935711127050846045, −5.54633344390425667840264944717, −4.94178959045265490855925218119, −4.30876348263583370054341056486, −4.06853526969133775380481934010, −4.02943360239990204590868483715, −2.89069247004404218362813725589, −2.56579511387070637701763941501, −1.88291078226632566944676563473, −1.50414249428293699898402646599, −0.54339113468104978389815767993, 0.54339113468104978389815767993, 1.50414249428293699898402646599, 1.88291078226632566944676563473, 2.56579511387070637701763941501, 2.89069247004404218362813725589, 4.02943360239990204590868483715, 4.06853526969133775380481934010, 4.30876348263583370054341056486, 4.94178959045265490855925218119, 5.54633344390425667840264944717, 5.87072247714935711127050846045, 6.14704461923121500029274219723, 6.82433041023246349280288512703, 7.23902651652528035790225865926, 7.39154135809763517554922142448, 7.946775133717328822803951042244, 8.291998783204522234294349930134, 8.785723803933609650617371643745, 8.886822846202992888782878355008, 9.672732593597194950984025557283

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