Properties

Label 4-790272-1.1-c1e2-0-35
Degree $4$
Conductor $790272$
Sign $1$
Analytic cond. $50.3884$
Root an. cond. $2.66429$
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  + 8·5-s − 7-s + 9-s + 4·11-s − 12·13-s + 38·25-s − 8·35-s − 8·43-s + 8·45-s + 24·47-s + 49-s + 32·55-s + 12·61-s − 63-s − 96·65-s − 16·67-s − 4·77-s + 81-s + 12·91-s + 4·99-s + 32·101-s − 32·103-s + 36·107-s + 20·113-s − 12·117-s − 10·121-s + 136·125-s + ⋯
L(s)  = 1  + 3.57·5-s − 0.377·7-s + 1/3·9-s + 1.20·11-s − 3.32·13-s + 38/5·25-s − 1.35·35-s − 1.21·43-s + 1.19·45-s + 3.50·47-s + 1/7·49-s + 4.31·55-s + 1.53·61-s − 0.125·63-s − 11.9·65-s − 1.95·67-s − 0.455·77-s + 1/9·81-s + 1.25·91-s + 0.402·99-s + 3.18·101-s − 3.15·103-s + 3.48·107-s + 1.88·113-s − 1.10·117-s − 0.909·121-s + 12.1·125-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 790272 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 790272 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(790272\)    =    \(2^{8} \cdot 3^{2} \cdot 7^{3}\)
Sign: $1$
Analytic conductor: \(50.3884\)
Root analytic conductor: \(2.66429\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 790272,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.043907586\)
\(L(\frac12)\) \(\approx\) \(4.043907586\)
\(L(\frac{3}{2})\) 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_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
7$C_1$ \( 1 + T \)
good5$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
11$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
23$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
29$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
31$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
41$C_2$ \( ( 1 + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
61$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
73$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
89$C_2$ \( ( 1 + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
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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

−8.557246747639740629686095607145, −7.48055140088893310165370888362, −7.16129745430315884399569933958, −6.96207355783957131001717701468, −6.29720922441833452882432131874, −6.05700480174216171276936327443, −5.57231195325264994616573262274, −5.14056436828631480262325484346, −4.80916909916633799382690392302, −4.24974945413085103790315507050, −3.18166849287238476303908403717, −2.53627159495471071974969043910, −2.24239796255929817450421188994, −1.86729050078829565643852869820, −0.995983380836290695648376877184, 0.995983380836290695648376877184, 1.86729050078829565643852869820, 2.24239796255929817450421188994, 2.53627159495471071974969043910, 3.18166849287238476303908403717, 4.24974945413085103790315507050, 4.80916909916633799382690392302, 5.14056436828631480262325484346, 5.57231195325264994616573262274, 6.05700480174216171276936327443, 6.29720922441833452882432131874, 6.96207355783957131001717701468, 7.16129745430315884399569933958, 7.48055140088893310165370888362, 8.557246747639740629686095607145

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