Properties

Label 4-252e2-1.1-c1e2-0-10
Degree $4$
Conductor $63504$
Sign $1$
Analytic cond. $4.04907$
Root an. cond. $1.41853$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·3-s − 2·5-s − 5·7-s + 6·9-s − 4·11-s − 3·13-s + 6·15-s − 7·17-s − 5·19-s + 15·21-s − 4·23-s + 5·25-s − 9·27-s + 29-s − 6·31-s + 12·33-s + 10·35-s − 11·37-s + 9·39-s + 9·41-s − 5·43-s − 12·45-s + 6·47-s + 18·49-s + 21·51-s − 3·53-s + 8·55-s + ⋯
L(s)  = 1  − 1.73·3-s − 0.894·5-s − 1.88·7-s + 2·9-s − 1.20·11-s − 0.832·13-s + 1.54·15-s − 1.69·17-s − 1.14·19-s + 3.27·21-s − 0.834·23-s + 25-s − 1.73·27-s + 0.185·29-s − 1.07·31-s + 2.08·33-s + 1.69·35-s − 1.80·37-s + 1.44·39-s + 1.40·41-s − 0.762·43-s − 1.78·45-s + 0.875·47-s + 18/7·49-s + 2.94·51-s − 0.412·53-s + 1.07·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63504 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63504 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(63504\)    =    \(2^{4} \cdot 3^{4} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(4.04907\)
Root analytic conductor: \(1.41853\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 63504,\ (\ :1/2, 1/2),\ 1)\)

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)$
bad2 \( 1 \)
3$C_2$ \( 1 + p T + p T^{2} \)
7$C_2$ \( 1 + 5 T + p T^{2} \)
good5$C_2^2$ \( 1 + 2 T - T^{2} + 2 p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + 4 T + 5 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
13$C_2^2$ \( 1 + 3 T - 4 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
17$C_2^2$ \( 1 + 7 T + 32 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 + 5 T + 6 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 + 4 T - 7 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - T - 28 T^{2} - p T^{3} + p^{2} T^{4} \)
31$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 + T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
41$C_2^2$ \( 1 - 9 T + 40 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
43$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 13 T + p T^{2} ) \)
47$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
53$C_2^2$ \( 1 + 3 T - 44 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 13 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 17 T + p T^{2} ) \)
79$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 + T - 82 T^{2} + p T^{3} + p^{2} T^{4} \)
89$C_2^2$ \( 1 + 15 T + 136 T^{2} + 15 p T^{3} + p^{2} T^{4} \)
97$C_2^2$ \( 1 - 17 T + 192 T^{2} - 17 p T^{3} + p^{2} T^{4} \)
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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

−11.91531417605403419779691848026, −11.23687913171523506384582319772, −10.83591144998887698239228116061, −10.55689329101202657812793391340, −10.05121210160052439283654383345, −9.662878219913839207319557658412, −8.850537160039160845037015473376, −8.552443764793535707587050143900, −7.47382598407465557345766399459, −7.29112617787487007250299519865, −6.55768890581043077752145308703, −6.46454446132880625459248694753, −5.64055223211726024280230055273, −5.21690857840584288507882888627, −4.35617907475976882257230554471, −4.10171835893774217139521115615, −3.08224443917003121759885613474, −2.23495678738140070114362636140, 0, 0, 2.23495678738140070114362636140, 3.08224443917003121759885613474, 4.10171835893774217139521115615, 4.35617907475976882257230554471, 5.21690857840584288507882888627, 5.64055223211726024280230055273, 6.46454446132880625459248694753, 6.55768890581043077752145308703, 7.29112617787487007250299519865, 7.47382598407465557345766399459, 8.552443764793535707587050143900, 8.850537160039160845037015473376, 9.662878219913839207319557658412, 10.05121210160052439283654383345, 10.55689329101202657812793391340, 10.83591144998887698239228116061, 11.23687913171523506384582319772, 11.91531417605403419779691848026

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