Properties

Label 4-1152e2-1.1-c1e2-0-26
Degree $4$
Conductor $1327104$
Sign $1$
Analytic cond. $84.6173$
Root an. cond. $3.03294$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·5-s − 4·17-s + 6·25-s + 12·29-s + 8·37-s + 12·41-s + 6·49-s − 4·53-s − 8·61-s + 12·73-s − 16·85-s + 12·89-s − 12·97-s − 20·101-s − 16·109-s + 12·113-s − 6·121-s + 4·125-s + 127-s + 131-s + 137-s + 139-s + 48·145-s + 149-s + 151-s + 157-s + 163-s + ⋯
L(s)  = 1  + 1.78·5-s − 0.970·17-s + 6/5·25-s + 2.22·29-s + 1.31·37-s + 1.87·41-s + 6/7·49-s − 0.549·53-s − 1.02·61-s + 1.40·73-s − 1.73·85-s + 1.27·89-s − 1.21·97-s − 1.99·101-s − 1.53·109-s + 1.12·113-s − 0.545·121-s + 0.357·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 3.98·145-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1327104 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1327104 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1327104\)    =    \(2^{14} \cdot 3^{4}\)
Sign: $1$
Analytic conductor: \(84.6173\)
Root analytic conductor: \(3.03294\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1327104,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.270380936\)
\(L(\frac12)\) \(\approx\) \(3.270380936\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good5$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \)
7$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
17$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
23$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
29$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 - 4 T + p T^{2} ) \)
31$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
37$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \)
41$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 - 2 T + p T^{2} ) \)
43$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
47$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
53$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \)
59$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \)
61$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
67$C_2^2$ \( 1 - 106 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
73$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
83$C_2^2$ \( 1 - 42 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
97$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
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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

−7.979822184315644601074690354491, −7.60711213609016019390932746453, −6.89428147945714149120611445805, −6.55637368973365420600646234300, −6.25155870829034170968935151784, −5.83764150913119265842724579563, −5.44275647464844509570306042402, −4.87202358930774992415886450435, −4.41084544944720286910182395374, −4.03880294192179075233581712915, −3.09032425339641161943213573635, −2.55491231538600778905395488855, −2.31590422558728039186678407727, −1.53184360943521357375879358282, −0.824898995986696111855477348977, 0.824898995986696111855477348977, 1.53184360943521357375879358282, 2.31590422558728039186678407727, 2.55491231538600778905395488855, 3.09032425339641161943213573635, 4.03880294192179075233581712915, 4.41084544944720286910182395374, 4.87202358930774992415886450435, 5.44275647464844509570306042402, 5.83764150913119265842724579563, 6.25155870829034170968935151784, 6.55637368973365420600646234300, 6.89428147945714149120611445805, 7.60711213609016019390932746453, 7.979822184315644601074690354491

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