Properties

Label 4-363e2-1.1-c1e2-0-0
Degree $4$
Conductor $131769$
Sign $1$
Analytic cond. $8.40170$
Root an. cond. $1.70251$
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  − 2·3-s − 3·4-s + 9-s + 6·12-s + 5·16-s + 2·25-s + 4·27-s + 4·31-s − 3·36-s − 16·37-s − 10·48-s − 12·49-s − 3·64-s − 4·67-s − 4·75-s − 11·81-s − 8·93-s − 4·97-s − 6·100-s + 16·103-s − 12·108-s + 32·111-s − 12·124-s + 127-s + 131-s + 137-s + 139-s + ⋯
L(s)  = 1  − 1.15·3-s − 3/2·4-s + 1/3·9-s + 1.73·12-s + 5/4·16-s + 2/5·25-s + 0.769·27-s + 0.718·31-s − 1/2·36-s − 2.63·37-s − 1.44·48-s − 1.71·49-s − 3/8·64-s − 0.488·67-s − 0.461·75-s − 1.22·81-s − 0.829·93-s − 0.406·97-s − 3/5·100-s + 1.57·103-s − 1.15·108-s + 3.03·111-s − 1.07·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 131769 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 131769 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(131769\)    =    \(3^{2} \cdot 11^{4}\)
Sign: $1$
Analytic conductor: \(8.40170\)
Root analytic conductor: \(1.70251\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 131769,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4284234086\)
\(L(\frac12)\) \(\approx\) \(0.4284234086\)
\(L(\frac{3}{2})\) 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)$
bad3$C_2$ \( 1 + 2 T + p T^{2} \)
11 \( 1 \)
good2$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
5$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
7$C_2^2$ \( 1 + 12 T^{2} + p^{2} T^{4} \)
13$C_2^2$ \( 1 + 8 T^{2} + p^{2} T^{4} \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
19$C_2^2$ \( 1 + 20 T^{2} + p^{2} T^{4} \)
23$C_2$ \( ( 1 - p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
31$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
43$C_2^2$ \( 1 + 68 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 86 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 74 T^{2} + p^{2} T^{4} \)
59$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 24 T^{2} + p^{2} T^{4} \)
67$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
71$C_2^2$ \( 1 - 134 T^{2} + p^{2} T^{4} \)
73$C_2^2$ \( 1 + 144 T^{2} + p^{2} T^{4} \)
79$C_2^2$ \( 1 + 140 T^{2} + p^{2} T^{4} \)
83$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
89$C_2$ \( ( 1 - p T^{2} )^{2} \)
97$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
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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

−9.405671286740997184554288495992, −8.797000790029441355982958488998, −8.579674991398873617309784797020, −8.057016631930084761513968154539, −7.39664373082666410736966671800, −6.70794367417227402791062043088, −6.41148018427408347594151303777, −5.59325352762443268171598514570, −5.32162302916273169093846107527, −4.75225035259969606442285663060, −4.42330448993699560487681894901, −3.59787274541159227045684373179, −3.01113778450263520241990964232, −1.67335836471364506317732969120, −0.50405745521329467502889003460, 0.50405745521329467502889003460, 1.67335836471364506317732969120, 3.01113778450263520241990964232, 3.59787274541159227045684373179, 4.42330448993699560487681894901, 4.75225035259969606442285663060, 5.32162302916273169093846107527, 5.59325352762443268171598514570, 6.41148018427408347594151303777, 6.70794367417227402791062043088, 7.39664373082666410736966671800, 8.057016631930084761513968154539, 8.579674991398873617309784797020, 8.797000790029441355982958488998, 9.405671286740997184554288495992

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