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Properties

Label 4-1040e2-1.1-c1e2-0-36
Degree $4$
Conductor $1081600$
Sign $1$
Analytic cond. $68.9637$
Root an. cond. $2.88174$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·5-s + 2·9-s + 6·13-s − 25-s + 12·29-s + 12·37-s − 4·45-s + 16·47-s − 14·49-s + 12·61-s − 12·65-s − 24·67-s − 12·73-s − 5·81-s + 8·83-s − 12·97-s + 12·101-s + 12·117-s + 18·121-s + 12·125-s + 127-s + 131-s + 137-s + 139-s − 24·145-s + 149-s + 151-s + ⋯
L(s)  = 1  − 0.894·5-s + 2/3·9-s + 1.66·13-s − 1/5·25-s + 2.22·29-s + 1.97·37-s − 0.596·45-s + 2.33·47-s − 2·49-s + 1.53·61-s − 1.48·65-s − 2.93·67-s − 1.40·73-s − 5/9·81-s + 0.878·83-s − 1.21·97-s + 1.19·101-s + 1.10·117-s + 1.63·121-s + 1.07·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 1.99·145-s + 0.0819·149-s + 0.0813·151-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1081600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1081600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1081600\)    =    \(2^{8} \cdot 5^{2} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(68.9637\)
Root analytic conductor: \(2.88174\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1081600,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.144180845\)
\(L(\frac12)\) \(\approx\) \(2.144180845\)
\(L(\frac{3}{2})\) 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)$Isogeny Class over $\mathbf{F}_p$
bad2 \( 1 \)
5$C_2$ \( 1 + 2 T + p T^{2} \)
13$C_2$ \( 1 - 6 T + p T^{2} \)
good3$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.3.a_ac
7$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.7.a_o
11$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \) 2.11.a_as
17$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.17.a_abi
19$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.19.a_ac
23$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.23.a_ak
29$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.29.am_dq
31$C_2^2$ \( 1 - 26 T^{2} + p^{2} T^{4} \) 2.31.a_aba
37$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.37.am_eg
41$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.41.a_as
43$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \) 2.43.a_aby
47$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.47.aq_gc
53$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \) 2.53.a_bm
59$C_2^2$ \( 1 - 114 T^{2} + p^{2} T^{4} \) 2.59.a_aek
61$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.61.am_gc
67$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.67.y_ks
71$C_2^2$ \( 1 - 138 T^{2} + p^{2} T^{4} \) 2.71.a_afi
73$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.73.m_ha
79$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.79.a_gc
83$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.83.ai_ha
89$C_2^2$ \( 1 - 114 T^{2} + p^{2} T^{4} \) 2.89.a_aek
97$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.97.m_iw
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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

−10.00173840385083004115063785330, −9.943309187719180617495087199600, −9.185556359148498521764584426874, −8.835489011682261773493672843094, −8.464388885528411953800306968009, −8.051035329127673152906327712165, −7.72557261735984306194659785749, −7.25089135135114879183420652461, −6.78552918420920999589288777055, −6.31426835738730583367383529068, −5.93797241021902873688716389666, −5.53993655651831529876263753437, −4.63330733158840173947606055483, −4.26546857643922934173021688713, −4.21884196437809164034625151005, −3.25963735609721207630069688768, −3.09176903652393932978968905610, −2.20106980713180414349391187764, −1.32405409542213580406814219862, −0.76070552880741975672395170803, 0.76070552880741975672395170803, 1.32405409542213580406814219862, 2.20106980713180414349391187764, 3.09176903652393932978968905610, 3.25963735609721207630069688768, 4.21884196437809164034625151005, 4.26546857643922934173021688713, 4.63330733158840173947606055483, 5.53993655651831529876263753437, 5.93797241021902873688716389666, 6.31426835738730583367383529068, 6.78552918420920999589288777055, 7.25089135135114879183420652461, 7.72557261735984306194659785749, 8.051035329127673152906327712165, 8.464388885528411953800306968009, 8.835489011682261773493672843094, 9.185556359148498521764584426874, 9.943309187719180617495087199600, 10.00173840385083004115063785330

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