Properties

Label 4-3360e2-1.1-c1e2-0-1
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  + 2·5-s − 9-s − 12·11-s + 4·19-s − 25-s + 20·29-s − 4·31-s − 12·41-s − 2·45-s − 49-s − 24·55-s + 8·59-s − 4·61-s − 20·71-s − 16·79-s + 81-s − 12·89-s + 8·95-s + 12·99-s − 12·101-s + 28·109-s + 86·121-s − 12·125-s + 127-s + 131-s + 137-s + 139-s + ⋯
L(s)  = 1  + 0.894·5-s − 1/3·9-s − 3.61·11-s + 0.917·19-s − 1/5·25-s + 3.71·29-s − 0.718·31-s − 1.87·41-s − 0.298·45-s − 1/7·49-s − 3.23·55-s + 1.04·59-s − 0.512·61-s − 2.37·71-s − 1.80·79-s + 1/9·81-s − 1.27·89-s + 0.820·95-s + 1.20·99-s − 1.19·101-s + 2.68·109-s + 7.81·121-s − 1.07·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-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

\(L(1)\) \(\approx\) \(0.9690264030\)
\(L(\frac12)\) \(\approx\) \(0.9690264030\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - 2 T + p T^{2} \)
7$C_2$ \( 1 + T^{2} \)
good11$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
17$C_2$ \( ( 1 - p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 + 58 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
79$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - p T^{2} )^{2} \)
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
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

−9.068146146729537114272930467029, −8.453199198244193886678299260433, −8.100198088762637971247728741586, −7.78013129243434975082598804130, −7.32889798147739994935466139538, −7.00786101523006531554763313414, −6.61330454811213264582109507050, −5.92179092235838098545030041937, −5.77498393794199755421930674200, −5.44236370150368742515792824190, −5.01251073711345163935679115923, −4.73215144914997529827776690150, −4.45376067954748012345844059185, −3.45400992377609536384687406305, −3.07507896326722737879049119379, −2.64803013172638326691747812633, −2.58150475567522762056755070276, −1.86726545830743380500800015427, −1.20126884126363522542248350808, −0.29949943485577545580429800866, 0.29949943485577545580429800866, 1.20126884126363522542248350808, 1.86726545830743380500800015427, 2.58150475567522762056755070276, 2.64803013172638326691747812633, 3.07507896326722737879049119379, 3.45400992377609536384687406305, 4.45376067954748012345844059185, 4.73215144914997529827776690150, 5.01251073711345163935679115923, 5.44236370150368742515792824190, 5.77498393794199755421930674200, 5.92179092235838098545030041937, 6.61330454811213264582109507050, 7.00786101523006531554763313414, 7.32889798147739994935466139538, 7.78013129243434975082598804130, 8.100198088762637971247728741586, 8.453199198244193886678299260433, 9.068146146729537114272930467029

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