Properties

Label 4-360e2-1.1-c1e2-0-31
Degree $4$
Conductor $129600$
Sign $1$
Analytic cond. $8.26340$
Root an. cond. $1.69546$
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  + 2·5-s + 8·11-s + 8·19-s − 25-s + 4·29-s − 4·41-s + 10·49-s + 16·55-s − 24·59-s − 20·61-s − 16·71-s − 32·79-s + 12·89-s + 16·95-s − 12·101-s + 12·109-s + 26·121-s − 12·125-s + 127-s + 131-s + 137-s + 139-s + 8·145-s + 149-s + 151-s + 157-s + 163-s + ⋯
L(s)  = 1  + 0.894·5-s + 2.41·11-s + 1.83·19-s − 1/5·25-s + 0.742·29-s − 0.624·41-s + 10/7·49-s + 2.15·55-s − 3.12·59-s − 2.56·61-s − 1.89·71-s − 3.60·79-s + 1.27·89-s + 1.64·95-s − 1.19·101-s + 1.14·109-s + 2.36·121-s − 1.07·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.664·145-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 129600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 129600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(129600\)    =    \(2^{6} \cdot 3^{4} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(8.26340\)
Root analytic conductor: \(1.69546\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{360} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 129600,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.296633300\)
\(L(\frac12)\) \(\approx\) \(2.296633300\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - 2 T + p T^{2} \)
good7$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
17$C_2$ \( ( 1 - p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 42 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
59$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 + 62 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 82 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + 16 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 162 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 + 62 T^{2} + p^{2} T^{4} \)
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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

−11.62146706578789122244066694284, −11.45985362797018267468215694097, −10.60889806535686417703199682791, −10.30702392713714639753250898354, −9.683191293094794001797824238203, −9.396936007551214622678105567950, −8.962362085644337362446530658841, −8.743542844396008738709760611756, −7.78163306240406971436674874828, −7.42096671005332491124093741311, −6.84725379222898699696651587228, −6.35013980618183755690568839468, −5.87617307431092642396527985062, −5.54004595599785564699608818615, −4.50006638623930153797112900260, −4.34534625404125172218814009652, −3.32051266049004067327964353717, −2.97156067584272290999047438192, −1.66107779292882608091093468010, −1.30437555056029370875722557072, 1.30437555056029370875722557072, 1.66107779292882608091093468010, 2.97156067584272290999047438192, 3.32051266049004067327964353717, 4.34534625404125172218814009652, 4.50006638623930153797112900260, 5.54004595599785564699608818615, 5.87617307431092642396527985062, 6.35013980618183755690568839468, 6.84725379222898699696651587228, 7.42096671005332491124093741311, 7.78163306240406971436674874828, 8.743542844396008738709760611756, 8.962362085644337362446530658841, 9.396936007551214622678105567950, 9.683191293094794001797824238203, 10.30702392713714639753250898354, 10.60889806535686417703199682791, 11.45985362797018267468215694097, 11.62146706578789122244066694284

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