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Properties

Label 4-266240-1.1-c1e2-0-4
Degree $4$
Conductor $266240$
Sign $1$
Analytic cond. $16.9756$
Root an. cond. $2.02981$
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  − 3·5-s + 2·9-s + 5·13-s + 8·17-s + 2·25-s + 4·29-s + 4·37-s + 8·41-s − 6·45-s − 10·49-s − 4·53-s + 4·61-s − 15·65-s + 12·73-s − 5·81-s − 24·85-s + 4·89-s − 24·97-s + 4·101-s + 10·117-s − 6·121-s + 10·125-s + 127-s + 131-s + 137-s + 139-s − 12·145-s + ⋯
L(s)  = 1  − 1.34·5-s + 2/3·9-s + 1.38·13-s + 1.94·17-s + 2/5·25-s + 0.742·29-s + 0.657·37-s + 1.24·41-s − 0.894·45-s − 1.42·49-s − 0.549·53-s + 0.512·61-s − 1.86·65-s + 1.40·73-s − 5/9·81-s − 2.60·85-s + 0.423·89-s − 2.43·97-s + 0.398·101-s + 0.924·117-s − 0.545·121-s + 0.894·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 0.996·145-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 266240 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 266240 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(266240\)    =    \(2^{12} \cdot 5 \cdot 13\)
Sign: $1$
Analytic conductor: \(16.9756\)
Root analytic conductor: \(2.02981\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 266240,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.627380842\)
\(L(\frac12)\) \(\approx\) \(1.627380842\)
\(L(\frac{3}{2})\) 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_1$$\times$$C_2$ \( ( 1 + T )( 1 + 2 T + p T^{2} ) \)
13$C_1$$\times$$C_2$ \( ( 1 + T )( 1 - 6 T + p T^{2} ) \)
good3$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.3.a_ac
7$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.7.a_k
11$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.11.a_g
17$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.17.ai_bu
19$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.19.a_ac
23$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.23.a_as
29$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.29.ae_ac
31$C_2^2$ \( 1 + 22 T^{2} + p^{2} T^{4} \) 2.31.a_w
37$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.37.ae_o
41$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.41.ai_dq
43$C_2^2$ \( 1 + 54 T^{2} + p^{2} T^{4} \) 2.43.a_cc
47$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.47.a_k
53$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.53.e_dq
59$C_2^2$ \( 1 + 62 T^{2} + p^{2} T^{4} \) 2.59.a_ck
61$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.61.ae_ck
67$C_2^2$ \( 1 - 110 T^{2} + p^{2} T^{4} \) 2.67.a_aeg
71$C_2^2$ \( 1 + 22 T^{2} + p^{2} T^{4} \) 2.71.a_w
73$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.73.am_gk
79$C_2^2$ \( 1 - 66 T^{2} + p^{2} T^{4} \) 2.79.a_aco
83$C_2^2$ \( 1 + 58 T^{2} + p^{2} T^{4} \) 2.83.a_cg
89$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.89.ae_eo
97$C_2$$\times$$C_2$ \( ( 1 + 10 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.97.y_mw
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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

−8.749488481082096603086787223146, −8.287824381754269075417376926614, −7.944865928037056618205494584812, −7.63823795008224470770331695042, −7.17976515743562206218631132929, −6.45074736110316133410167152812, −6.17492654025570442904489961273, −5.45895224957657440509867817043, −4.96969863257109238936941031514, −4.15374578902191187015177805685, −3.95631593856304328259266259179, −3.35062476844355781546456513255, −2.80982636668835691483274875577, −1.55509454879015341938421947632, −0.851866183428669626803672058593, 0.851866183428669626803672058593, 1.55509454879015341938421947632, 2.80982636668835691483274875577, 3.35062476844355781546456513255, 3.95631593856304328259266259179, 4.15374578902191187015177805685, 4.96969863257109238936941031514, 5.45895224957657440509867817043, 6.17492654025570442904489961273, 6.45074736110316133410167152812, 7.17976515743562206218631132929, 7.63823795008224470770331695042, 7.944865928037056618205494584812, 8.287824381754269075417376926614, 8.749488481082096603086787223146

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