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Properties

Label 4-2e18-1.1-c1e2-0-24
Degree $4$
Conductor $262144$
Sign $1$
Analytic cond. $16.7145$
Root an. cond. $2.02196$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 8·7-s − 4·9-s − 8·17-s − 8·23-s − 2·25-s − 16·31-s + 4·41-s + 34·49-s + 32·63-s + 8·71-s − 8·73-s + 16·79-s + 7·81-s + 24·89-s − 8·97-s − 24·103-s + 28·113-s + 64·119-s − 20·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 32·153-s + 157-s + ⋯
L(s)  = 1  − 3.02·7-s − 4/3·9-s − 1.94·17-s − 1.66·23-s − 2/5·25-s − 2.87·31-s + 0.624·41-s + 34/7·49-s + 4.03·63-s + 0.949·71-s − 0.936·73-s + 1.80·79-s + 7/9·81-s + 2.54·89-s − 0.812·97-s − 2.36·103-s + 2.63·113-s + 5.86·119-s − 1.81·121-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 + 2.58·153-s + 0.0798·157-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 262144 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 262144 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(262144\)    =    \(2^{18}\)
Sign: $1$
Analytic conductor: \(16.7145\)
Root analytic conductor: \(2.02196\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 262144,\ (\ :1/2, 1/2),\ 1)\)

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 \( 1 \)
good3$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \) 2.3.a_e
5$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.5.a_c
7$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.7.i_be
11$C_2^2$ \( 1 + 20 T^{2} + p^{2} T^{4} \) 2.11.a_u
13$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \) 2.13.a_s
17$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.17.i_by
19$C_2^2$ \( 1 - 12 T^{2} + p^{2} T^{4} \) 2.19.a_am
23$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.23.i_ck
29$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \) 2.29.a_ao
31$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.31.q_ew
37$C_2^2$ \( 1 + 66 T^{2} + p^{2} T^{4} \) 2.37.a_co
41$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.41.ae_di
43$C_2^2$ \( 1 + 68 T^{2} + p^{2} T^{4} \) 2.43.a_cq
47$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.47.a_dq
53$C_2^2$ \( 1 + 98 T^{2} + p^{2} T^{4} \) 2.53.a_du
59$C_2^2$ \( 1 + 100 T^{2} + p^{2} T^{4} \) 2.59.a_dw
61$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \) 2.61.a_by
67$C_2^2$ \( 1 + 116 T^{2} + p^{2} T^{4} \) 2.67.a_em
71$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.71.ai_gc
73$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.73.i_gg
79$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.79.aq_io
83$C_2^2$ \( 1 + 68 T^{2} + p^{2} T^{4} \) 2.83.a_cq
89$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \) 2.89.ay_mk
97$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.97.i_ic
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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

−10.49574187132408215607831995199, −10.33304860268379518842665720076, −9.614809267164176818448192764351, −9.281525011802992645370023439182, −9.112097050998776594121264527480, −8.686721910286336691620343293808, −7.945112543853197062870817498787, −7.46622527459382883180435886894, −6.77474245661909789851894232306, −6.52878082543246757169759583565, −6.08610173063719762075708845260, −5.76660807984295845765798898658, −5.17422502549788900099779959683, −4.16725586767261259610490344282, −3.64602790789663783204629244220, −3.41070935135273440025139991518, −2.51934096660172210523376964460, −2.20310925372371644223791846829, 0, 0, 2.20310925372371644223791846829, 2.51934096660172210523376964460, 3.41070935135273440025139991518, 3.64602790789663783204629244220, 4.16725586767261259610490344282, 5.17422502549788900099779959683, 5.76660807984295845765798898658, 6.08610173063719762075708845260, 6.52878082543246757169759583565, 6.77474245661909789851894232306, 7.46622527459382883180435886894, 7.945112543853197062870817498787, 8.686721910286336691620343293808, 9.112097050998776594121264527480, 9.281525011802992645370023439182, 9.614809267164176818448192764351, 10.33304860268379518842665720076, 10.49574187132408215607831995199

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