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Properties

Label 4-462e2-1.1-c1e2-0-21
Degree $4$
Conductor $213444$
Sign $-1$
Analytic cond. $13.6093$
Root an. cond. $1.92070$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $1$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s + 4-s − 2·7-s + 9-s + 2·12-s − 8·13-s + 16-s + 8·19-s − 4·21-s − 6·25-s − 4·27-s − 2·28-s − 20·31-s + 36-s − 12·37-s − 16·39-s − 8·43-s + 2·48-s + 3·49-s − 8·52-s + 16·57-s − 16·61-s − 2·63-s + 64-s + 16·67-s + 8·73-s − 12·75-s + ⋯
L(s)  = 1  + 1.15·3-s + 1/2·4-s − 0.755·7-s + 1/3·9-s + 0.577·12-s − 2.21·13-s + 1/4·16-s + 1.83·19-s − 0.872·21-s − 6/5·25-s − 0.769·27-s − 0.377·28-s − 3.59·31-s + 1/6·36-s − 1.97·37-s − 2.56·39-s − 1.21·43-s + 0.288·48-s + 3/7·49-s − 1.10·52-s + 2.11·57-s − 2.04·61-s − 0.251·63-s + 1/8·64-s + 1.95·67-s + 0.936·73-s − 1.38·75-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 213444 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 213444 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(213444\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2} \cdot 11^{2}\)
Sign: $-1$
Analytic conductor: \(13.6093\)
Root analytic conductor: \(1.92070\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 213444,\ (\ :1/2, 1/2),\ -1)\)

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$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
3$C_2$ \( 1 - 2 T + p T^{2} \)
7$C_1$ \( ( 1 + T )^{2} \)
11$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.5.a_g
13$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.13.i_bq
17$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.17.a_bi
19$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.19.ai_cc
23$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.23.a_be
29$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.29.a_cc
31$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.31.u_gg
37$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.37.m_eg
41$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.41.a_de
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.43.i_dy
47$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.47.a_ag
53$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.53.a_adm
59$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.59.a_s
61$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.61.q_he
67$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.67.aq_hq
71$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.71.a_ew
73$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.73.ai_gg
79$C_2$ \( ( 1 - 16 T + p T^{2} )^{2} \) 2.79.abg_py
83$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.83.a_fu
89$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.89.a_da
97$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.97.am_iw
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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

−9.240853610425596933131644882823, −8.085838808436017661666180332862, −7.87359639769148641923907597256, −7.47160076799445304063607589326, −6.89070428269537648461178127009, −6.81376239244202158997461277739, −5.64283316816673405468499778451, −5.41745538950503066981292745262, −4.96360142275870366848623108181, −3.90068439822755419122695958551, −3.29966281651344024951011938340, −3.23324115647284242077206608800, −2.06919705445307860507213607024, −2.00597606234881909225076458985, 0, 2.00597606234881909225076458985, 2.06919705445307860507213607024, 3.23324115647284242077206608800, 3.29966281651344024951011938340, 3.90068439822755419122695958551, 4.96360142275870366848623108181, 5.41745538950503066981292745262, 5.64283316816673405468499778451, 6.81376239244202158997461277739, 6.89070428269537648461178127009, 7.47160076799445304063607589326, 7.87359639769148641923907597256, 8.085838808436017661666180332862, 9.240853610425596933131644882823

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