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Properties

Label 4-266240-1.1-c1e2-0-21
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5-s + 2·9-s − 7·13-s − 4·17-s − 2·25-s − 4·29-s + 4·37-s − 12·41-s + 2·45-s + 2·49-s + 4·53-s + 12·61-s − 7·65-s + 4·73-s − 5·81-s − 4·85-s − 20·89-s + 20·97-s − 12·101-s + 4·109-s − 12·113-s − 14·117-s − 14·121-s − 10·125-s + 127-s + 131-s + 137-s + ⋯
L(s)  = 1  + 0.447·5-s + 2/3·9-s − 1.94·13-s − 0.970·17-s − 2/5·25-s − 0.742·29-s + 0.657·37-s − 1.87·41-s + 0.298·45-s + 2/7·49-s + 0.549·53-s + 1.53·61-s − 0.868·65-s + 0.468·73-s − 5/9·81-s − 0.433·85-s − 2.11·89-s + 2.03·97-s − 1.19·101-s + 0.383·109-s − 1.12·113-s − 1.29·117-s − 1.27·121-s − 0.894·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-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: \(1\)
Selberg data: \((4,\ 266240,\ (\ :1/2, 1/2),\ -1)\)

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)$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$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.7.a_ac
11$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.11.a_o
17$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.17.e_w
19$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.19.a_o
23$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.23.a_ac
29$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.29.e_bu
31$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \) 2.31.a_abi
37$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.37.ae_o
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.41.m_eo
43$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \) 2.43.a_as
47$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \) 2.47.a_as
53$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.53.ae_bu
59$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \) 2.59.a_be
61$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.61.am_fm
67$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) 2.67.a_aes
71$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \) 2.71.a_afa
73$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.73.ae_di
79$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) 2.79.a_adu
83$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.83.a_w
89$C_2$$\times$$C_2$ \( ( 1 + 6 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.89.u_kc
97$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.97.au_li
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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

−8.758050296511726230700068534058, −8.216096133241165749980515794971, −7.51264639445207250348636410233, −7.39015491758430470082512799442, −6.69312974568368747364832000088, −6.51073153163966418050185507620, −5.65558085709889602867179242981, −5.18237366590523089688750568150, −4.82358174542458678483469086761, −4.14424196332518117454177597162, −3.68438279428901387251886744353, −2.62859896772525413794083972559, −2.31216829056906708213700401955, −1.49645039747710750855212974684, 0, 1.49645039747710750855212974684, 2.31216829056906708213700401955, 2.62859896772525413794083972559, 3.68438279428901387251886744353, 4.14424196332518117454177597162, 4.82358174542458678483469086761, 5.18237366590523089688750568150, 5.65558085709889602867179242981, 6.51073153163966418050185507620, 6.69312974568368747364832000088, 7.39015491758430470082512799442, 7.51264639445207250348636410233, 8.216096133241165749980515794971, 8.758050296511726230700068534058

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