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Properties

Label 4-9360e2-1.1-c1e2-0-19
Degree $4$
Conductor $87609600$
Sign $1$
Analytic cond. $5586.06$
Root an. cond. $8.64522$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·5-s + 6·11-s − 2·13-s − 4·17-s − 6·19-s − 2·23-s + 3·25-s − 8·29-s − 2·31-s − 8·37-s + 4·41-s + 2·43-s + 8·47-s − 14·49-s + 4·53-s + 12·55-s + 2·59-s − 4·65-s − 12·67-s − 6·71-s − 24·73-s − 4·79-s + 8·83-s − 8·85-s − 20·89-s − 12·95-s − 20·97-s + ⋯
L(s)  = 1  + 0.894·5-s + 1.80·11-s − 0.554·13-s − 0.970·17-s − 1.37·19-s − 0.417·23-s + 3/5·25-s − 1.48·29-s − 0.359·31-s − 1.31·37-s + 0.624·41-s + 0.304·43-s + 1.16·47-s − 2·49-s + 0.549·53-s + 1.61·55-s + 0.260·59-s − 0.496·65-s − 1.46·67-s − 0.712·71-s − 2.80·73-s − 0.450·79-s + 0.878·83-s − 0.867·85-s − 2.11·89-s − 1.23·95-s − 2.03·97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 87609600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 87609600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(87609600\)    =    \(2^{8} \cdot 3^{4} \cdot 5^{2} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(5586.06\)
Root analytic conductor: \(8.64522\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 87609600,\ (\ :1/2, 1/2),\ 1)\)

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 \( 1 \)
5$C_1$ \( ( 1 - T )^{2} \)
13$C_1$ \( ( 1 + T )^{2} \)
good7$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.7.a_o
11$C_4$ \( 1 - 6 T + 26 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.11.ag_ba
17$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.17.e_bm
19$D_{4}$ \( 1 + 6 T + 42 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.19.g_bq
23$D_{4}$ \( 1 + 2 T + 42 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.23.c_bq
29$C_4$ \( 1 + 8 T + 54 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.29.i_cc
31$D_{4}$ \( 1 + 2 T + 18 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.31.c_s
37$D_{4}$ \( 1 + 8 T + 70 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.37.i_cs
41$C_4$ \( 1 - 4 T + 6 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.41.ae_g
43$D_{4}$ \( 1 - 2 T + 82 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.43.ac_de
47$D_{4}$ \( 1 - 8 T + 30 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.47.ai_be
53$D_{4}$ \( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.53.ae_be
59$D_{4}$ \( 1 - 2 T + 74 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.59.ac_cw
61$C_2^2$ \( 1 + 102 T^{2} + p^{2} T^{4} \) 2.61.a_dy
67$D_{4}$ \( 1 + 12 T + 150 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.67.m_fu
71$D_{4}$ \( 1 + 6 T + 146 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.71.g_fq
73$D_{4}$ \( 1 + 24 T + 270 T^{2} + 24 p T^{3} + p^{2} T^{4} \) 2.73.y_kk
79$D_{4}$ \( 1 + 4 T + 142 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.79.e_fm
83$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.83.ai_ha
89$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.89.u_ks
97$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.97.u_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

−7.32528465495988304631423541998, −7.14716017863086900023140237417, −6.72522284600748965856639734297, −6.66389075693166651223389235451, −6.15761272881055029899608734207, −5.85826773704523250404020558786, −5.52689218429173939399842681449, −5.31074688833152813403754924142, −4.52044002506552924633637335638, −4.43216327111233501983180769240, −4.00267098349224159635694869177, −3.89337311645652559953517630512, −3.07294820340271518977377721450, −2.90714493868132380909683568780, −2.20238782359475020367649432397, −2.00313350973763790314008066535, −1.42806010816876280071723649462, −1.31640764684290981613069667801, 0, 0, 1.31640764684290981613069667801, 1.42806010816876280071723649462, 2.00313350973763790314008066535, 2.20238782359475020367649432397, 2.90714493868132380909683568780, 3.07294820340271518977377721450, 3.89337311645652559953517630512, 4.00267098349224159635694869177, 4.43216327111233501983180769240, 4.52044002506552924633637335638, 5.31074688833152813403754924142, 5.52689218429173939399842681449, 5.85826773704523250404020558786, 6.15761272881055029899608734207, 6.66389075693166651223389235451, 6.72522284600748965856639734297, 7.14716017863086900023140237417, 7.32528465495988304631423541998

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