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Properties

Label 4-2e18-1.1-c1e2-0-7
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 $0$

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: \(0\)
Selberg data: \((4,\ 262144,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.297820698\)
\(L(\frac12)\) \(\approx\) \(2.297820698\)
\(L(\frac{3}{2})\) 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.ai_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.ai_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.aq_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.i_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.q_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

−11.16864900599586194902434800352, −10.91209404595769029210519404696, −10.42637178033083658237172163282, −9.883314653812860460528287596748, −8.936781326194524102910309356880, −8.813439232030150242587399693514, −8.487612586958332116151505354964, −8.167058260342329260108230696044, −7.41511515077245583457173870416, −7.39445193201164554674056408186, −6.24521647187241645309584533271, −6.23489109353507760925346992833, −5.20880474696415961785815476276, −5.02210215319635069469174047645, −4.49960645185009216472236312771, −4.23899392513018733142803174210, −3.00436965059298781127362087828, −2.48954608600981803648358007327, −1.85171982909904310984154993292, −0.976243472162187210614835272268, 0.976243472162187210614835272268, 1.85171982909904310984154993292, 2.48954608600981803648358007327, 3.00436965059298781127362087828, 4.23899392513018733142803174210, 4.49960645185009216472236312771, 5.02210215319635069469174047645, 5.20880474696415961785815476276, 6.23489109353507760925346992833, 6.24521647187241645309584533271, 7.39445193201164554674056408186, 7.41511515077245583457173870416, 8.167058260342329260108230696044, 8.487612586958332116151505354964, 8.813439232030150242587399693514, 8.936781326194524102910309356880, 9.883314653812860460528287596748, 10.42637178033083658237172163282, 10.91209404595769029210519404696, 11.16864900599586194902434800352

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