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Properties

Label 4-5760e2-1.1-c1e2-0-23
Degree $4$
Conductor $33177600$
Sign $1$
Analytic cond. $2115.43$
Root an. cond. $6.78187$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·5-s + 2·7-s + 4·11-s − 2·13-s − 6·17-s − 2·19-s + 2·23-s + 3·25-s − 6·31-s + 4·35-s + 2·37-s + 4·41-s − 4·43-s + 18·47-s + 6·49-s + 20·53-s + 8·55-s + 12·59-s + 4·61-s − 4·65-s − 4·67-s + 16·71-s + 8·73-s + 8·77-s − 18·79-s − 8·83-s − 12·85-s + ⋯
L(s)  = 1  + 0.894·5-s + 0.755·7-s + 1.20·11-s − 0.554·13-s − 1.45·17-s − 0.458·19-s + 0.417·23-s + 3/5·25-s − 1.07·31-s + 0.676·35-s + 0.328·37-s + 0.624·41-s − 0.609·43-s + 2.62·47-s + 6/7·49-s + 2.74·53-s + 1.07·55-s + 1.56·59-s + 0.512·61-s − 0.496·65-s − 0.488·67-s + 1.89·71-s + 0.936·73-s + 0.911·77-s − 2.02·79-s − 0.878·83-s − 1.30·85-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 33177600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33177600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(33177600\)    =    \(2^{14} \cdot 3^{4} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(2115.43\)
Root analytic conductor: \(6.78187\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 33177600,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.990844827\)
\(L(\frac12)\) \(\approx\) \(4.990844827\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{2} \)
good7$D_{4}$ \( 1 - 2 T - 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.7.ac_ac
11$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.11.ae_ba
13$C_4$ \( 1 + 2 T + 10 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.13.c_k
17$D_{4}$ \( 1 + 6 T + 26 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.17.g_ba
19$D_{4}$ \( 1 + 2 T + 22 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.19.c_w
23$D_{4}$ \( 1 - 2 T + 30 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.23.ac_be
29$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.29.a_ak
31$D_{4}$ \( 1 + 6 T + 54 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.31.g_cc
37$D_{4}$ \( 1 - 2 T + 58 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.37.ac_cg
41$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.41.ae_di
43$D_{4}$ \( 1 + 4 T + 22 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.43.e_w
47$D_{4}$ \( 1 - 18 T + 158 T^{2} - 18 p T^{3} + p^{2} T^{4} \) 2.47.as_gc
53$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.53.au_hy
59$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.59.am_fy
61$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.61.ae_ew
67$D_{4}$ \( 1 + 4 T + 70 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.67.e_cs
71$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.71.aq_hy
73$D_{4}$ \( 1 - 8 T + 94 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.73.ai_dq
79$D_{4}$ \( 1 + 18 T + 222 T^{2} + 18 p T^{3} + p^{2} T^{4} \) 2.79.s_io
83$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.83.i_ha
89$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.89.au_ks
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

−8.335222533329505734184525348322, −8.001644622231715371082345259487, −7.43286128492331394867783341088, −7.16906061895871221893224221668, −6.91970303859033905154740306463, −6.51081298161083353454745227353, −6.26777697646044694182577333275, −5.67011872211686513557467980031, −5.33631576215886632835076523247, −5.31698707969663180318281023552, −4.47703493935471993178056520398, −4.36716286697370194724504853527, −3.92245559356035957251808557618, −3.64356268505293763576523808570, −2.83680163156455920787425347615, −2.41364692880760398643922920785, −2.00563147968681177008788447608, −1.90534242605803672747254329121, −0.918468970184361290672860681967, −0.70474660435550256179888916351, 0.70474660435550256179888916351, 0.918468970184361290672860681967, 1.90534242605803672747254329121, 2.00563147968681177008788447608, 2.41364692880760398643922920785, 2.83680163156455920787425347615, 3.64356268505293763576523808570, 3.92245559356035957251808557618, 4.36716286697370194724504853527, 4.47703493935471993178056520398, 5.31698707969663180318281023552, 5.33631576215886632835076523247, 5.67011872211686513557467980031, 6.26777697646044694182577333275, 6.51081298161083353454745227353, 6.91970303859033905154740306463, 7.16906061895871221893224221668, 7.43286128492331394867783341088, 8.001644622231715371082345259487, 8.335222533329505734184525348322

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