Properties

Label 4-3360e2-1.1-c1e2-0-7
Degree $4$
Conductor $11289600$
Sign $1$
Analytic cond. $719.834$
Root an. cond. $5.17974$
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 − 9-s + 8·11-s + 12·19-s + 11·25-s + 12·29-s − 4·31-s + 24·41-s + 4·45-s − 49-s − 32·55-s − 24·59-s − 20·61-s + 16·71-s + 16·79-s + 81-s + 24·89-s − 48·95-s − 8·99-s − 20·109-s + 26·121-s − 24·125-s + 127-s + 131-s + 137-s + 139-s − 48·145-s + ⋯
L(s)  = 1  − 1.78·5-s − 1/3·9-s + 2.41·11-s + 2.75·19-s + 11/5·25-s + 2.22·29-s − 0.718·31-s + 3.74·41-s + 0.596·45-s − 1/7·49-s − 4.31·55-s − 3.12·59-s − 2.56·61-s + 1.89·71-s + 1.80·79-s + 1/9·81-s + 2.54·89-s − 4.92·95-s − 0.804·99-s − 1.91·109-s + 2.36·121-s − 2.14·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 3.98·145-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 11289600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11289600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(11289600\)    =    \(2^{10} \cdot 3^{2} \cdot 5^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(719.834\)
Root analytic conductor: \(5.17974\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{3360} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 11289600,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.602249051\)
\(L(\frac12)\) \(\approx\) \(2.602249051\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T^{2} \)
5$C_2$ \( 1 + 4 T + p T^{2} \)
7$C_2$ \( 1 + T^{2} \)
good11$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
13$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 42 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
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$C_2^2$ \( 1 - 102 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 118 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 142 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - p T^{2} )^{2} \)
89$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 - 94 T^{2} + p^{2} T^{4} \)
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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

−9.090278864648666465341224446487, −8.218989491862241874310563928510, −8.064299758356909863675232984616, −7.55288325098702594165502106098, −7.51226570486987670801716930340, −7.10805706004313871067638185457, −6.45812107844451681419746120773, −6.26122938090300512744682722937, −6.04780182219825164866892677847, −5.23231614210226916337449333408, −4.84014988807504099471872870284, −4.56893854666655810046859401015, −4.09016668267637507937135721745, −3.55966027357382780450031188523, −3.55108049990210595287526687804, −2.89422433923791605270181446126, −2.56027479478351291638804228502, −1.33082579950994232297089636410, −1.17505176232683934023706049905, −0.61356664369414126430990298668, 0.61356664369414126430990298668, 1.17505176232683934023706049905, 1.33082579950994232297089636410, 2.56027479478351291638804228502, 2.89422433923791605270181446126, 3.55108049990210595287526687804, 3.55966027357382780450031188523, 4.09016668267637507937135721745, 4.56893854666655810046859401015, 4.84014988807504099471872870284, 5.23231614210226916337449333408, 6.04780182219825164866892677847, 6.26122938090300512744682722937, 6.45812107844451681419746120773, 7.10805706004313871067638185457, 7.51226570486987670801716930340, 7.55288325098702594165502106098, 8.064299758356909863675232984616, 8.218989491862241874310563928510, 9.090278864648666465341224446487

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