Properties

Label 4-2880e2-1.1-c1e2-0-18
Degree $4$
Conductor $8294400$
Sign $1$
Analytic cond. $528.858$
Root an. cond. $4.79550$
Motivic weight $1$
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  − 4·5-s − 4·11-s + 11·25-s + 16·31-s − 4·41-s + 10·49-s + 16·55-s + 20·59-s − 4·61-s + 24·71-s − 20·89-s − 16·101-s + 20·109-s − 10·121-s − 24·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 64·155-s + 157-s + 163-s + 167-s − 10·169-s + 173-s + ⋯
L(s)  = 1  − 1.78·5-s − 1.20·11-s + 11/5·25-s + 2.87·31-s − 0.624·41-s + 10/7·49-s + 2.15·55-s + 2.60·59-s − 0.512·61-s + 2.84·71-s − 2.11·89-s − 1.59·101-s + 1.91·109-s − 0.909·121-s − 2.14·125-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 − 5.14·155-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 0.769·169-s + 0.0760·173-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8294400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8294400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(8294400\)    =    \(2^{12} \cdot 3^{4} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(528.858\)
Root analytic conductor: \(4.79550\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 8294400,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.650949424\)
\(L(\frac12)\) \(\approx\) \(1.650949424\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5$C_2$ \( 1 + 4 T + p T^{2} \)
good7$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
19$C_2$ \( ( 1 + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
41$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 150 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \)
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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

−8.594191784662286254895475544511, −8.365953026244954835700443483040, −8.307863491848735583883189139518, −8.001574168889058733595011254681, −7.45738866666933489167923488435, −7.13246809260482024570078023922, −6.76380566577230156510053940932, −6.54465596499595912921405750865, −5.75514649364597399498908434869, −5.50012948324977129225454404235, −4.99307484288483297737515179756, −4.63120421854270456717038520802, −4.09898264355439309626318444353, −4.04104180268986066270647567450, −3.20331205634706938569931812347, −3.01086302087051235545026328621, −2.51161396654636142003547732614, −1.90590387885158983943837332724, −0.73243203376052582314454589771, −0.64885553122997107544489675342, 0.64885553122997107544489675342, 0.73243203376052582314454589771, 1.90590387885158983943837332724, 2.51161396654636142003547732614, 3.01086302087051235545026328621, 3.20331205634706938569931812347, 4.04104180268986066270647567450, 4.09898264355439309626318444353, 4.63120421854270456717038520802, 4.99307484288483297737515179756, 5.50012948324977129225454404235, 5.75514649364597399498908434869, 6.54465596499595912921405750865, 6.76380566577230156510053940932, 7.13246809260482024570078023922, 7.45738866666933489167923488435, 8.001574168889058733595011254681, 8.307863491848735583883189139518, 8.365953026244954835700443483040, 8.594191784662286254895475544511

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