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Properties

Label 4-55e4-1.1-c1e2-0-5
Degree $4$
Conductor $9150625$
Sign $1$
Analytic cond. $583.451$
Root an. cond. $4.91474$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·3-s − 4-s + 6·9-s + 4·12-s − 3·16-s − 12·23-s + 4·27-s + 8·31-s − 6·36-s − 20·37-s + 12·47-s + 12·48-s − 2·49-s + 12·53-s + 7·64-s − 20·67-s + 48·69-s − 37·81-s − 12·89-s + 12·92-s − 32·93-s + 20·97-s − 4·103-s − 4·108-s + 80·111-s − 12·113-s − 8·124-s + ⋯
L(s)  = 1  − 2.30·3-s − 1/2·4-s + 2·9-s + 1.15·12-s − 3/4·16-s − 2.50·23-s + 0.769·27-s + 1.43·31-s − 36-s − 3.28·37-s + 1.75·47-s + 1.73·48-s − 2/7·49-s + 1.64·53-s + 7/8·64-s − 2.44·67-s + 5.77·69-s − 4.11·81-s − 1.27·89-s + 1.25·92-s − 3.31·93-s + 2.03·97-s − 0.394·103-s − 0.384·108-s + 7.59·111-s − 1.12·113-s − 0.718·124-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9150625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9150625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(9150625\)    =    \(5^{4} \cdot 11^{4}\)
Sign: $1$
Analytic conductor: \(583.451\)
Root analytic conductor: \(4.91474\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 9150625,\ (\ :1/2, 1/2),\ 1)\)

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 $\mathbf{F}_p$
bad5 \( 1 \)
11 \( 1 \)
good2$C_2^2$ \( 1 + T^{2} + p^{2} T^{4} \) 2.2.a_b
3$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.3.e_k
7$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.7.a_c
13$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.13.a_ba
17$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \) 2.17.a_ao
19$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.19.a_ak
23$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.23.m_de
29$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.29.a_cg
31$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.31.ai_da
37$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.37.u_gs
41$C_2^2$ \( 1 + 34 T^{2} + p^{2} T^{4} \) 2.41.a_bi
43$C_2^2$ \( 1 + 74 T^{2} + p^{2} T^{4} \) 2.43.a_cw
47$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.47.am_fa
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.53.am_fm
59$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.59.a_eo
61$C_2^2$ \( 1 + 74 T^{2} + p^{2} T^{4} \) 2.61.a_cw
67$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.67.u_ja
71$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.71.a_fm
73$C_2^2$ \( 1 + 98 T^{2} + p^{2} T^{4} \) 2.73.a_du
79$C_2^2$ \( 1 + 110 T^{2} + p^{2} T^{4} \) 2.79.a_eg
83$C_2^2$ \( 1 - 134 T^{2} + p^{2} T^{4} \) 2.83.a_afe
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.89.m_ig
97$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.97.au_li
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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

−8.459739175974775755095470422068, −8.366233693642661372013880388151, −7.65681961315103489242083808402, −7.28084559401010097618895313607, −6.84271947968059261852205481523, −6.54471988288324646127848098086, −6.16292638385766678195213580494, −5.76380550638857076846614545834, −5.59407683641837506880540688727, −5.17437584526464953764901082216, −4.67889960416948878982598264478, −4.48912506605511150287815189496, −3.93988461365387891873046069700, −3.55456899226672612462503650411, −2.75626991463482279653623135928, −2.27207608553518378978334225758, −1.58699312365382491621906581844, −0.888542465546367366742737291825, 0, 0, 0.888542465546367366742737291825, 1.58699312365382491621906581844, 2.27207608553518378978334225758, 2.75626991463482279653623135928, 3.55456899226672612462503650411, 3.93988461365387891873046069700, 4.48912506605511150287815189496, 4.67889960416948878982598264478, 5.17437584526464953764901082216, 5.59407683641837506880540688727, 5.76380550638857076846614545834, 6.16292638385766678195213580494, 6.54471988288324646127848098086, 6.84271947968059261852205481523, 7.28084559401010097618895313607, 7.65681961315103489242083808402, 8.366233693642661372013880388151, 8.459739175974775755095470422068

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