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Properties

Label 4-5200e2-1.1-c1e2-0-10
Degree $4$
Conductor $27040000$
Sign $1$
Analytic cond. $1724.09$
Root an. cond. $6.44377$
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  + 2·7-s + 2·9-s − 10·11-s − 2·13-s + 2·17-s − 4·19-s − 12·23-s − 6·29-s + 6·31-s + 12·37-s − 4·43-s + 10·47-s − 9·49-s − 6·53-s − 18·59-s + 2·61-s + 4·63-s − 14·67-s − 4·71-s + 12·73-s − 20·77-s + 12·79-s − 5·81-s − 6·83-s − 4·91-s − 4·97-s − 20·99-s + ⋯
L(s)  = 1  + 0.755·7-s + 2/3·9-s − 3.01·11-s − 0.554·13-s + 0.485·17-s − 0.917·19-s − 2.50·23-s − 1.11·29-s + 1.07·31-s + 1.97·37-s − 0.609·43-s + 1.45·47-s − 9/7·49-s − 0.824·53-s − 2.34·59-s + 0.256·61-s + 0.503·63-s − 1.71·67-s − 0.474·71-s + 1.40·73-s − 2.27·77-s + 1.35·79-s − 5/9·81-s − 0.658·83-s − 0.419·91-s − 0.406·97-s − 2.01·99-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 27040000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27040000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(27040000\)    =    \(2^{8} \cdot 5^{4} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(1724.09\)
Root analytic conductor: \(6.44377\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 27040000,\ (\ :1/2, 1/2),\ 1)\)

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 \( 1 \)
5 \( 1 \)
13$C_1$ \( ( 1 + T )^{2} \)
good3$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.3.a_ac
7$D_{4}$ \( 1 - 2 T + 13 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.7.ac_n
11$D_{4}$ \( 1 + 10 T + 45 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.11.k_bt
17$D_{4}$ \( 1 - 2 T + 27 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.17.ac_bb
19$D_{4}$ \( 1 + 4 T + 10 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.19.e_k
23$C_4$ \( 1 + 12 T + 74 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.23.m_cw
29$D_{4}$ \( 1 + 6 T + 59 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.29.g_ch
31$D_{4}$ \( 1 - 6 T + 53 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.31.ag_cb
37$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.37.am_eg
41$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \) 2.41.a_by
43$D_{4}$ \( 1 + 4 T + 82 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.43.e_de
47$D_{4}$ \( 1 - 10 T + 117 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.47.ak_en
53$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.53.g_el
59$D_{4}$ \( 1 + 18 T + 181 T^{2} + 18 p T^{3} + p^{2} T^{4} \) 2.59.s_gz
61$C_2$ \( ( 1 - T + p T^{2} )^{2} \) 2.61.ac_et
67$D_{4}$ \( 1 + 14 T + 165 T^{2} + 14 p T^{3} + p^{2} T^{4} \) 2.67.o_gj
71$D_{4}$ \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.71.e_s
73$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.73.am_ha
79$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.79.am_hm
83$D_{4}$ \( 1 + 6 T + 77 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.83.g_cz
89$C_2^2$ \( 1 + 106 T^{2} + p^{2} T^{4} \) 2.89.a_ec
97$D_{4}$ \( 1 + 4 T + 190 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.97.e_hi
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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

−7.973024707113002633250041498116, −7.76487488690859993728907296826, −7.48586602779685670934103311151, −7.21432148116673124204116557767, −6.47957452984299973677379446285, −6.14665508285740862660902882036, −5.80075240942158368708670426241, −5.57439570463457864857147194808, −5.03527450821182605482583200771, −4.60547225597599437710922647273, −4.50277916125585323095520526599, −4.14661316805055466607982984603, −3.32759580232836193051167688042, −3.09863537135794772428400501996, −2.41111168646470400253053170534, −2.20701742635764125632291699499, −1.85361404647478438906837681904, −1.09121918988343824635096263529, 0, 0, 1.09121918988343824635096263529, 1.85361404647478438906837681904, 2.20701742635764125632291699499, 2.41111168646470400253053170534, 3.09863537135794772428400501996, 3.32759580232836193051167688042, 4.14661316805055466607982984603, 4.50277916125585323095520526599, 4.60547225597599437710922647273, 5.03527450821182605482583200771, 5.57439570463457864857147194808, 5.80075240942158368708670426241, 6.14665508285740862660902882036, 6.47957452984299973677379446285, 7.21432148116673124204116557767, 7.48586602779685670934103311151, 7.76487488690859993728907296826, 7.973024707113002633250041498116

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