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Properties

Label 4-855e2-1.1-c1e2-0-16
Degree $4$
Conductor $731025$
Sign $1$
Analytic cond. $46.6107$
Root an. cond. $2.61289$
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  − 2·2-s + 4-s + 2·5-s − 4·10-s − 4·11-s − 4·13-s + 16-s − 8·17-s − 2·19-s + 2·20-s + 8·22-s − 4·23-s + 3·25-s + 8·26-s − 12·31-s + 2·32-s + 16·34-s − 4·37-s + 4·38-s − 8·41-s + 16·43-s − 4·44-s + 8·46-s + 4·47-s − 12·49-s − 6·50-s − 4·52-s + ⋯
L(s)  = 1  − 1.41·2-s + 1/2·4-s + 0.894·5-s − 1.26·10-s − 1.20·11-s − 1.10·13-s + 1/4·16-s − 1.94·17-s − 0.458·19-s + 0.447·20-s + 1.70·22-s − 0.834·23-s + 3/5·25-s + 1.56·26-s − 2.15·31-s + 0.353·32-s + 2.74·34-s − 0.657·37-s + 0.648·38-s − 1.24·41-s + 2.43·43-s − 0.603·44-s + 1.17·46-s + 0.583·47-s − 1.71·49-s − 0.848·50-s − 0.554·52-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 731025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(731025\)    =    \(3^{4} \cdot 5^{2} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(46.6107\)
Root analytic conductor: \(2.61289\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 731025,\ (\ :1/2, 1/2),\ 1)\)

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 $\mathbf{F}_p$
bad3 \( 1 \)
5$C_1$ \( ( 1 - T )^{2} \)
19$C_1$ \( ( 1 + T )^{2} \)
good2$D_{4}$ \( 1 + p T + 3 T^{2} + p^{2} T^{3} + p^{2} T^{4} \) 2.2.c_d
7$C_2^2$ \( 1 + 12 T^{2} + p^{2} T^{4} \) 2.7.a_m
11$C_2^2$ \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.11.e_i
13$D_{4}$ \( 1 + 4 T + 28 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.13.e_bc
17$D_{4}$ \( 1 + 8 T + 42 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.17.i_bq
23$D_{4}$ \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.23.e_s
29$C_2^2$ \( 1 + 56 T^{2} + p^{2} T^{4} \) 2.29.a_ce
31$D_{4}$ \( 1 + 12 T + 90 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.31.m_dm
37$D_{4}$ \( 1 + 4 T + 76 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.37.e_cy
41$D_{4}$ \( 1 + 8 T + 80 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.41.i_dc
43$D_{4}$ \( 1 - 16 T + 132 T^{2} - 16 p T^{3} + p^{2} T^{4} \) 2.43.aq_fc
47$D_{4}$ \( 1 - 4 T + 66 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.47.ae_co
53$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.53.q_go
59$D_{4}$ \( 1 + 8 T + 62 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.59.i_ck
61$D_{4}$ \( 1 + 8 T + 10 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.61.i_k
67$D_{4}$ \( 1 + 8 T + 118 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.67.i_eo
71$D_{4}$ \( 1 - 16 T + 198 T^{2} - 16 p T^{3} + p^{2} T^{4} \) 2.71.aq_hq
73$D_{4}$ \( 1 + 4 T + 118 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.73.e_eo
79$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.79.a_gc
83$D_{4}$ \( 1 + 20 T + 258 T^{2} + 20 p T^{3} + p^{2} T^{4} \) 2.83.u_jy
89$D_{4}$ \( 1 - 8 T + 96 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.89.ai_ds
97$D_{4}$ \( 1 + 28 T + 372 T^{2} + 28 p T^{3} + p^{2} T^{4} \) 2.97.bc_oi
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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

−9.722745648379049918646211507621, −9.514531458906954383141396676411, −9.227281111221382192476111583475, −8.866871035962801030941946926719, −8.214413845173848281060607772219, −8.187536557251497965678579957603, −7.37133136499828317476175292112, −7.25258869154653640498590323148, −6.59797370219109302594742518380, −6.04254267696844185646324438513, −5.76376858388568670659294503497, −5.03561476211193453184203592506, −4.69800476778969242459409458966, −4.19892764820537397559298863593, −3.24134624149838719801586204400, −2.66266741777911759740479304679, −2.06707568249052481512345127651, −1.65084733236590813945970926393, 0, 0, 1.65084733236590813945970926393, 2.06707568249052481512345127651, 2.66266741777911759740479304679, 3.24134624149838719801586204400, 4.19892764820537397559298863593, 4.69800476778969242459409458966, 5.03561476211193453184203592506, 5.76376858388568670659294503497, 6.04254267696844185646324438513, 6.59797370219109302594742518380, 7.25258869154653640498590323148, 7.37133136499828317476175292112, 8.187536557251497965678579957603, 8.214413845173848281060607772219, 8.866871035962801030941946926719, 9.227281111221382192476111583475, 9.514531458906954383141396676411, 9.722745648379049918646211507621

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