Properties

Label 4-221184-1.1-c1e2-0-24
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 9-s + 4·11-s − 6·17-s − 6·25-s − 27-s − 4·33-s − 6·41-s − 12·43-s − 2·49-s + 6·51-s + 4·59-s − 4·73-s + 6·75-s + 81-s − 12·83-s − 30·89-s − 16·97-s + 4·99-s + 20·107-s + 18·113-s − 10·121-s + 6·123-s + ⋯
L(s)  = 1  − 0.577·3-s + 1/3·9-s + 1.20·11-s − 1.45·17-s − 6/5·25-s − 0.192·27-s − 0.696·33-s − 0.937·41-s − 1.82·43-s − 2/7·49-s + 0.840·51-s + 0.520·59-s − 0.468·73-s + 0.692·75-s + 1/9·81-s − 1.31·83-s − 3.17·89-s − 1.62·97-s + 0.402·99-s + 1.93·107-s + 1.69·113-s − 0.909·121-s + 0.541·123-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: \(1\)
Selberg data: \((4,\ 221184,\ (\ :1/2, 1/2),\ -1)\)

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 \( 1 \)
3$C_1$ \( 1 + T \)
good5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.5.a_g
7$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.7.a_c
11$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.11.ae_ba
13$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \) 2.13.a_ao
17$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.17.g_bq
19$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.19.a_w
23$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \) 2.23.a_abe
29$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.29.a_o
31$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.31.a_k
37$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.37.a_aba
41$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.41.g_de
43$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.43.m_di
47$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \) 2.47.a_abe
53$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \) 2.53.a_bm
59$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.59.ae_di
61$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.61.a_w
67$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.67.a_eo
71$C_2^2$ \( 1 + 34 T^{2} + p^{2} T^{4} \) 2.71.a_bi
73$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.73.e_g
79$C_2^2$ \( 1 - 46 T^{2} + p^{2} T^{4} \) 2.79.a_abu
83$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) 2.83.m_cg
89$C_2$$\times$$C_2$ \( ( 1 + 14 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) 2.89.be_pm
97$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.97.q_io
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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

−8.662365585640960441087254072603, −8.504458991608582144473696466841, −7.86746701221832268160287276949, −7.14544208370585086700448578579, −6.77467933923774534261444857084, −6.52689902424219843983209694809, −5.83386415434304320342692595658, −5.45456257988404177613840645772, −4.67027594438241727802122294649, −4.29266806836651780394996632519, −3.75574035252798841286365336149, −3.03762309827932559751420027552, −2.04457277768799908661913153089, −1.45265442644416162580974074768, 0, 1.45265442644416162580974074768, 2.04457277768799908661913153089, 3.03762309827932559751420027552, 3.75574035252798841286365336149, 4.29266806836651780394996632519, 4.67027594438241727802122294649, 5.45456257988404177613840645772, 5.83386415434304320342692595658, 6.52689902424219843983209694809, 6.77467933923774534261444857084, 7.14544208370585086700448578579, 7.86746701221832268160287276949, 8.504458991608582144473696466841, 8.662365585640960441087254072603

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