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Properties

Label 4-56e4-1.1-c1e2-0-23
Degree $4$
Conductor $9834496$
Sign $1$
Analytic cond. $627.055$
Root an. cond. $5.00410$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·9-s − 8·11-s − 2·25-s − 4·29-s − 20·37-s − 8·43-s − 12·53-s + 24·67-s − 16·79-s − 5·81-s − 16·99-s − 24·107-s − 4·109-s + 20·113-s + 26·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 18·169-s + 173-s + 179-s + ⋯
L(s)  = 1  + 2/3·9-s − 2.41·11-s − 2/5·25-s − 0.742·29-s − 3.28·37-s − 1.21·43-s − 1.64·53-s + 2.93·67-s − 1.80·79-s − 5/9·81-s − 1.60·99-s − 2.32·107-s − 0.383·109-s + 1.88·113-s + 2.36·121-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 + 0.0783·163-s + 0.0773·167-s − 1.38·169-s + 0.0760·173-s + 0.0747·179-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9834496 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9834496 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(9834496\)    =    \(2^{12} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(627.055\)
Root analytic conductor: \(5.00410\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 9834496,\ (\ :1/2, 1/2),\ 1)\)

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 \( 1 \)
7 \( 1 \)
good3$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.3.a_ac
5$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.5.a_c
11$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.11.i_bm
13$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \) 2.13.a_s
17$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.17.a_c
19$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \) 2.19.a_be
23$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.23.a_bu
29$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.29.e_ck
31$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \) 2.31.a_be
37$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.37.u_gs
41$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \) 2.41.a_by
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.43.i_dy
47$C_2^2$ \( 1 + 62 T^{2} + p^{2} T^{4} \) 2.47.a_ck
53$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.53.m_fm
59$C_2^2$ \( 1 + 110 T^{2} + p^{2} T^{4} \) 2.59.a_eg
61$C_2^2$ \( 1 - 78 T^{2} + p^{2} T^{4} \) 2.61.a_ada
67$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \) 2.67.ay_ks
71$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.71.a_fm
73$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.73.a_fq
79$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.79.q_io
83$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \) 2.83.a_abi
89$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.89.a_gw
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

−8.434520498618258128084172296659, −8.191042881382082392551978094233, −7.77671276225346516113571040562, −7.29361168457613831441277568071, −7.15918213822715456212138337468, −6.71970893035379121009720877414, −6.25637909547117129480387369248, −5.71319751657130402676879938331, −5.31619000780868682975687138472, −5.06517989502424964369175902724, −4.86431677435581061504887168048, −4.19019144479728860513926098522, −3.57509259917546250621529482784, −3.44917693916690013647662235062, −2.74640207550887347679169458151, −2.35456797228664862766580065174, −1.80167898873644632512393059908, −1.33298917568129315587963758165, 0, 0, 1.33298917568129315587963758165, 1.80167898873644632512393059908, 2.35456797228664862766580065174, 2.74640207550887347679169458151, 3.44917693916690013647662235062, 3.57509259917546250621529482784, 4.19019144479728860513926098522, 4.86431677435581061504887168048, 5.06517989502424964369175902724, 5.31619000780868682975687138472, 5.71319751657130402676879938331, 6.25637909547117129480387369248, 6.71970893035379121009720877414, 7.15918213822715456212138337468, 7.29361168457613831441277568071, 7.77671276225346516113571040562, 8.191042881382082392551978094233, 8.434520498618258128084172296659

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