Properties

Label 4-221184-1.1-c1e2-0-19
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

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 9-s − 4·11-s + 2·17-s + 2·25-s − 27-s + 4·33-s + 2·41-s − 12·43-s − 10·49-s − 2·51-s + 4·59-s − 16·67-s − 20·73-s − 2·75-s + 81-s + 12·83-s + 18·89-s − 4·99-s + 4·107-s + 18·113-s + 6·121-s − 2·123-s + ⋯
L(s)  = 1  − 0.577·3-s + 1/3·9-s − 1.20·11-s + 0.485·17-s + 2/5·25-s − 0.192·27-s + 0.696·33-s + 0.312·41-s − 1.82·43-s − 1.42·49-s − 0.280·51-s + 0.520·59-s − 1.95·67-s − 2.34·73-s − 0.230·75-s + 1/9·81-s + 1.31·83-s + 1.90·89-s − 0.402·99-s + 0.386·107-s + 1.69·113-s + 6/11·121-s − 0.180·123-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$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.5.a_ac
7$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.7.a_k
11$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.11.e_k
13$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \) 2.13.a_s
17$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.17.ac_k
19$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.19.a_w
23$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.23.a_c
29$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \) 2.29.a_bm
31$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \) 2.31.a_ao
37$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.37.a_ak
41$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + p T^{2} ) \) 2.41.ac_de
43$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.43.m_eo
47$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.47.a_c
53$C_2^2$ \( 1 + 62 T^{2} + p^{2} T^{4} \) 2.53.a_ck
59$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.59.ae_di
61$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \) 2.61.a_bm
67$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.67.q_ha
71$C_2^2$ \( 1 - 94 T^{2} + p^{2} T^{4} \) 2.71.a_adq
73$C_2$$\times$$C_2$ \( ( 1 + 6 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.73.u_iw
79$C_2^2$ \( 1 - 102 T^{2} + p^{2} T^{4} \) 2.79.a_ady
83$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.83.am_hu
89$C_2$$\times$$C_2$ \( ( 1 - 18 T + p T^{2} )( 1 + p T^{2} ) \) 2.89.as_gw
97$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.97.a_ac
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

−8.789875202163892998273778488883, −8.292718087157895392588884978432, −7.67544684941507091025515121933, −7.52728960436240235413745298919, −6.84295501051977109228196297265, −6.27892371405502939600549010618, −5.92176269475469027686636398524, −5.20432603966542850586531693221, −4.94160198230902232728094967409, −4.40210362196707086055765470843, −3.50131652578575223108645481593, −3.05251626949135318059511762107, −2.22695255909458503557873774718, −1.32729937706052229864836775055, 0, 1.32729937706052229864836775055, 2.22695255909458503557873774718, 3.05251626949135318059511762107, 3.50131652578575223108645481593, 4.40210362196707086055765470843, 4.94160198230902232728094967409, 5.20432603966542850586531693221, 5.92176269475469027686636398524, 6.27892371405502939600549010618, 6.84295501051977109228196297265, 7.52728960436240235413745298919, 7.67544684941507091025515121933, 8.292718087157895392588884978432, 8.789875202163892998273778488883

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