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Properties

Label 4-1344e2-1.1-c1e2-0-49
Degree $4$
Conductor $1806336$
Sign $-1$
Analytic cond. $115.173$
Root an. cond. $3.27595$
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  + 2·7-s + 9-s − 6·11-s − 4·23-s + 2·25-s − 4·37-s + 10·43-s − 3·49-s − 4·53-s + 2·63-s − 2·67-s + 4·71-s − 12·77-s + 24·79-s + 81-s − 6·99-s − 14·107-s − 20·109-s − 8·113-s + 14·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + ⋯
L(s)  = 1  + 0.755·7-s + 1/3·9-s − 1.80·11-s − 0.834·23-s + 2/5·25-s − 0.657·37-s + 1.52·43-s − 3/7·49-s − 0.549·53-s + 0.251·63-s − 0.244·67-s + 0.474·71-s − 1.36·77-s + 2.70·79-s + 1/9·81-s − 0.603·99-s − 1.35·107-s − 1.91·109-s − 0.752·113-s + 1.27·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1806336 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1806336 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1806336\)    =    \(2^{12} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-1$
Analytic conductor: \(115.173\)
Root analytic conductor: \(3.27595\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 1806336,\ (\ :1/2, 1/2),\ -1)\)

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 \( 1 \)
3$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
7$C_2$ \( 1 - 2 T + p T^{2} \)
good5$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.5.a_ac
11$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.11.g_w
13$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \) 2.13.a_g
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.17.a_ac
19$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \) 2.19.a_be
23$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.23.e_bi
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.29.a_w
31$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.31.a_c
37$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.37.e_ck
41$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.41.a_k
43$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) 2.43.ak_eg
47$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \) 2.47.a_by
53$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.53.e_abi
59$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \) 2.59.a_bm
61$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.61.a_o
67$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.67.c_cc
71$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.71.ae_bu
73$C_2^2$ \( 1 + 58 T^{2} + p^{2} T^{4} \) 2.73.a_cg
79$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 - 10 T + p T^{2} ) \) 2.79.ay_lm
83$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \) 2.83.a_acg
89$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.89.a_ac
97$C_2^2$ \( 1 + 162 T^{2} + p^{2} T^{4} \) 2.97.a_gg
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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

−7.73610067984365931712136403899, −7.34539009845465169741037329392, −6.61353696270975125558800128410, −6.50083634516695474374822313221, −5.60994725171421288900300643191, −5.49681646689936107560189184595, −4.99472193654022144603292181434, −4.57330484807354072464267120552, −4.09769701704306453206415727989, −3.49958218288214073989535878325, −2.88854994760096348327395621602, −2.34751045342836577111638319294, −1.90067392512467637747143960220, −1.04587928648818471612528834810, 0, 1.04587928648818471612528834810, 1.90067392512467637747143960220, 2.34751045342836577111638319294, 2.88854994760096348327395621602, 3.49958218288214073989535878325, 4.09769701704306453206415727989, 4.57330484807354072464267120552, 4.99472193654022144603292181434, 5.49681646689936107560189184595, 5.60994725171421288900300643191, 6.50083634516695474374822313221, 6.61353696270975125558800128410, 7.34539009845465169741037329392, 7.73610067984365931712136403899

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