Properties

Label 4-790272-1.1-c1e2-0-62
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 $1$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·5-s + 7-s − 3·9-s − 12·17-s + 2·25-s + 4·35-s − 4·37-s + 4·41-s + 8·43-s − 12·45-s + 16·47-s + 49-s − 3·63-s + 8·67-s − 32·79-s + 9·81-s − 16·83-s − 48·85-s − 12·89-s + 4·101-s − 20·109-s − 12·119-s − 6·121-s − 28·125-s + 127-s + 131-s + 137-s + ⋯
L(s)  = 1  + 1.78·5-s + 0.377·7-s − 9-s − 2.91·17-s + 2/5·25-s + 0.676·35-s − 0.657·37-s + 0.624·41-s + 1.21·43-s − 1.78·45-s + 2.33·47-s + 1/7·49-s − 0.377·63-s + 0.977·67-s − 3.60·79-s + 81-s − 1.75·83-s − 5.20·85-s − 1.27·89-s + 0.398·101-s − 1.91·109-s − 1.10·119-s − 0.545·121-s − 2.50·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-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: $\chi_{790272} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 790272,\ (\ :1/2, 1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + p T^{2} \)
7$C_1$ \( 1 - T \)
good5$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
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$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
37$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
73$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
79$C_2$ \( ( 1 + 16 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
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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

−8.257254837686236479707568177745, −7.48402841893068646023884414833, −7.02621174890446736965278831122, −6.67949194679031572918581545842, −6.02280276296176526986201378935, −5.82613210142107824134972732669, −5.51675247245265629993912731057, −4.88671808931178581792400546928, −4.18465932715971087267449515138, −4.05428052119787997110157361479, −2.83043555474359072971239832626, −2.43940180539462742301847104796, −2.14556791802447766709825303506, −1.37997143341888120500168617355, 0, 1.37997143341888120500168617355, 2.14556791802447766709825303506, 2.43940180539462742301847104796, 2.83043555474359072971239832626, 4.05428052119787997110157361479, 4.18465932715971087267449515138, 4.88671808931178581792400546928, 5.51675247245265629993912731057, 5.82613210142107824134972732669, 6.02280276296176526986201378935, 6.67949194679031572918581545842, 7.02621174890446736965278831122, 7.48402841893068646023884414833, 8.257254837686236479707568177745

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