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Properties

Label 4-3120e2-1.1-c0e2-0-6
Degree $4$
Conductor $9734400$
Sign $1$
Analytic cond. $2.42450$
Root an. cond. $1.24783$
Motivic weight $0$
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  + 2·3-s − 4-s + 2·5-s + 3·9-s − 2·11-s − 2·12-s + 4·15-s + 16-s − 2·20-s + 3·25-s + 4·27-s − 4·33-s − 3·36-s + 2·44-s + 6·45-s − 2·47-s + 2·48-s − 4·55-s − 2·59-s − 4·60-s + 2·61-s − 64-s + 6·75-s + 2·80-s + 5·81-s + 4·83-s − 6·99-s + ⋯
L(s)  = 1  + 2·3-s − 4-s + 2·5-s + 3·9-s − 2·11-s − 2·12-s + 4·15-s + 16-s − 2·20-s + 3·25-s + 4·27-s − 4·33-s − 3·36-s + 2·44-s + 6·45-s − 2·47-s + 2·48-s − 4·55-s − 2·59-s − 4·60-s + 2·61-s − 64-s + 6·75-s + 2·80-s + 5·81-s + 4·83-s − 6·99-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9734400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9734400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(9734400\)    =    \(2^{8} \cdot 3^{2} \cdot 5^{2} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(2.42450\)
Root analytic conductor: \(1.24783\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 9734400,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.401983035\)
\(L(\frac12)\) \(\approx\) \(3.401983035\)
\(L(1)\) 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$C_2$ \( 1 + T^{2} \)
3$C_1$ \( ( 1 - T )^{2} \)
5$C_1$ \( ( 1 - T )^{2} \)
13$C_2$ \( 1 + T^{2} \)
good7$C_2^2$ \( 1 + T^{4} \)
11$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
17$C_2^2$ \( 1 + T^{4} \)
19$C_2^2$ \( 1 + T^{4} \)
23$C_2^2$ \( 1 + T^{4} \)
29$C_2^2$ \( 1 + T^{4} \)
31$C_2$ \( ( 1 + T^{2} )^{2} \)
37$C_2$ \( ( 1 + T^{2} )^{2} \)
41$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
43$C_2$ \( ( 1 + T^{2} )^{2} \)
47$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
53$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
59$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
61$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
67$C_2$ \( ( 1 + T^{2} )^{2} \)
71$C_2$ \( ( 1 + T^{2} )^{2} \)
73$C_2^2$ \( 1 + T^{4} \)
79$C_2$ \( ( 1 + T^{2} )^{2} \)
83$C_1$ \( ( 1 - T )^{4} \)
89$C_2$ \( ( 1 + T^{2} )^{2} \)
97$C_2^2$ \( 1 + T^{4} \)
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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

−9.110000778761161927338246029646, −8.662619783091493859067188349189, −8.279278556162001924311673605817, −8.258048378412766658462105824666, −7.57732331310658119779307421080, −7.49544336034108061658455259527, −6.89481145635884210142070041791, −6.27535678684710522357357024314, −6.23594173499046276770766549568, −5.41349323283068553572615381465, −5.15952107035195224876390695669, −4.73741575966261569694415275689, −4.66987706946254784537436111372, −3.67382990029089376153461557608, −3.50830179108020617865847865999, −2.91449158227643011550084916224, −2.51361146193342594907370237039, −2.24244777211437787107058008926, −1.65175507396444904373510744622, −1.10242295363192986271165330554, 1.10242295363192986271165330554, 1.65175507396444904373510744622, 2.24244777211437787107058008926, 2.51361146193342594907370237039, 2.91449158227643011550084916224, 3.50830179108020617865847865999, 3.67382990029089376153461557608, 4.66987706946254784537436111372, 4.73741575966261569694415275689, 5.15952107035195224876390695669, 5.41349323283068553572615381465, 6.23594173499046276770766549568, 6.27535678684710522357357024314, 6.89481145635884210142070041791, 7.49544336034108061658455259527, 7.57732331310658119779307421080, 8.258048378412766658462105824666, 8.279278556162001924311673605817, 8.662619783091493859067188349189, 9.110000778761161927338246029646

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