Properties

Label 4-4608e2-1.1-c1e2-0-25
Degree $4$
Conductor $21233664$
Sign $1$
Analytic cond. $1353.87$
Root an. cond. $6.06589$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 12·11-s + 12·17-s + 8·19-s − 8·25-s + 4·41-s + 4·49-s − 8·59-s + 16·67-s + 16·73-s − 4·83-s − 4·89-s + 4·97-s + 8·107-s + 20·113-s + 86·121-s + ⋯
L(s)  = 1  − 3.61·11-s + 2.91·17-s + 1.83·19-s − 8/5·25-s + 0.624·41-s + 4/7·49-s − 1.04·59-s + 1.95·67-s + 1.87·73-s − 0.439·83-s − 0.423·89-s + 0.406·97-s + 0.773·107-s + 1.88·113-s + 7.81·121-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 21233664 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21233664 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(21233664\)    =    \(2^{18} \cdot 3^{4}\)
Sign: $1$
Analytic conductor: \(1353.87\)
Root analytic conductor: \(6.06589\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 21233664,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.306617593\)
\(L(\frac12)\) \(\approx\) \(2.306617593\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good5$C_2^2$ \( 1 + 8 T^{2} + p^{2} T^{4} \) 2.5.a_i
7$C_2^2$ \( 1 - 4 T^{2} + p^{2} T^{4} \) 2.7.a_ae
11$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.11.m_cg
13$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \) 2.13.a_ag
17$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.17.am_cs
19$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.19.ai_cc
23$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \) 2.23.a_bm
29$C_2^2$ \( 1 + 56 T^{2} + p^{2} T^{4} \) 2.29.a_ce
31$C_2^2$ \( 1 + 60 T^{2} + p^{2} T^{4} \) 2.31.a_ci
37$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.37.a_c
41$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.41.ae_di
43$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.43.a_di
47$C_2^2$ \( 1 + 86 T^{2} + p^{2} T^{4} \) 2.47.a_di
53$C_2^2$ \( 1 + 8 T^{2} + p^{2} T^{4} \) 2.53.a_i
59$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.59.i_fe
61$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \) 2.61.a_by
67$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.67.aq_hq
71$C_2^2$ \( 1 + 134 T^{2} + p^{2} T^{4} \) 2.71.a_fe
73$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.73.aq_ic
79$C_2^2$ \( 1 - 4 T^{2} + p^{2} T^{4} \) 2.79.a_ae
83$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.83.e_go
89$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.89.e_ha
97$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.97.ae_hq
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.276184087266556528535122354929, −8.009473957235000967330810873660, −7.70352746305477667480353965616, −7.63687164610730112986378270375, −7.28062220876485213823497085781, −6.89952722412933334158490184967, −5.98065861003891703916258301790, −5.79575944257530186179878689380, −5.49589258688188924039897851753, −5.41462593282243605796669919867, −4.81348933183792684835472464424, −4.70515435805137415906135440703, −3.73300385330888394602038847270, −3.50439550883325499933772449766, −3.00747817755691272887712103948, −2.85150440808542344203998586447, −2.21448598200301824713335395687, −1.80284531855813003661234071606, −0.866966124798289090124052354248, −0.53474195002049884930665648204, 0.53474195002049884930665648204, 0.866966124798289090124052354248, 1.80284531855813003661234071606, 2.21448598200301824713335395687, 2.85150440808542344203998586447, 3.00747817755691272887712103948, 3.50439550883325499933772449766, 3.73300385330888394602038847270, 4.70515435805137415906135440703, 4.81348933183792684835472464424, 5.41462593282243605796669919867, 5.49589258688188924039897851753, 5.79575944257530186179878689380, 5.98065861003891703916258301790, 6.89952722412933334158490184967, 7.28062220876485213823497085781, 7.63687164610730112986378270375, 7.70352746305477667480353965616, 8.009473957235000967330810873660, 8.276184087266556528535122354929

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