Properties

Label 4-221184-1.1-c1e2-0-4
Degree $4$
Conductor $221184$
Sign $1$
Analytic cond. $14.1028$
Root an. cond. $1.93788$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 9-s + 8·13-s − 8·23-s + 2·25-s − 27-s − 8·39-s + 8·47-s − 6·49-s + 8·59-s + 16·61-s + 8·69-s + 8·71-s + 4·73-s − 2·75-s + 81-s + 16·83-s − 12·97-s − 8·107-s − 8·109-s + 8·117-s − 22·121-s + ⋯
L(s)  = 1  − 0.577·3-s + 1/3·9-s + 2.21·13-s − 1.66·23-s + 2/5·25-s − 0.192·27-s − 1.28·39-s + 1.16·47-s − 6/7·49-s + 1.04·59-s + 2.04·61-s + 0.963·69-s + 0.949·71-s + 0.468·73-s − 0.230·75-s + 1/9·81-s + 1.75·83-s − 1.21·97-s − 0.773·107-s − 0.766·109-s + 0.739·117-s − 2·121-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 221184 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 221184 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(221184\)    =    \(2^{13} \cdot 3^{3}\)
Sign: $1$
Analytic conductor: \(14.1028\)
Root analytic conductor: \(1.93788\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 221184,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.505292890\)
\(L(\frac12)\) \(\approx\) \(1.505292890\)
\(L(\frac{3}{2})\) 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$C_1$ \( 1 + T \)
good5$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.5.a_ac
7$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \) 2.7.a_g
11$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.11.a_w
13$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.13.ai_bm
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.17.a_ac
19$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.19.a_aba
23$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.23.i_bu
29$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.29.a_ac
31$C_2^2$ \( 1 + 22 T^{2} + p^{2} T^{4} \) 2.31.a_w
37$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.37.a_cs
41$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.41.a_as
43$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.43.a_w
47$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) 2.47.ai_dq
53$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \) 2.53.a_as
59$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.59.ai_cs
61$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) 2.61.aq_ha
67$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.67.a_ak
71$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) 2.71.ai_fm
73$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.73.ae_di
79$C_2^2$ \( 1 - 106 T^{2} + p^{2} T^{4} \) 2.79.a_aec
83$C_2$$\times$$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + p T^{2} ) \) 2.83.aq_gk
89$C_2^2$ \( 1 + 110 T^{2} + p^{2} T^{4} \) 2.89.a_eg
97$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.97.m_gk
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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

−9.063990486462948824704899612401, −8.456406080154355445358005066204, −8.090256393654089561246836233355, −7.79520914947883523086396146226, −6.82542240502094288657576927247, −6.64996093657912203707435845685, −6.13960819477416781937898462973, −5.57245369354955348645891345343, −5.30069522161736473061297469588, −4.39227427151129716256641633278, −3.84215029261319003238684844443, −3.60492975137269895393158464159, −2.54771034548512628652568121915, −1.72128169505805947304696181080, −0.839159547994740327718557664273, 0.839159547994740327718557664273, 1.72128169505805947304696181080, 2.54771034548512628652568121915, 3.60492975137269895393158464159, 3.84215029261319003238684844443, 4.39227427151129716256641633278, 5.30069522161736473061297469588, 5.57245369354955348645891345343, 6.13960819477416781937898462973, 6.64996093657912203707435845685, 6.82542240502094288657576927247, 7.79520914947883523086396146226, 8.090256393654089561246836233355, 8.456406080154355445358005066204, 9.063990486462948824704899612401

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