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Properties

Label 4-285e2-1.1-c1e2-0-3
Degree $4$
Conductor $81225$
Sign $1$
Analytic cond. $5.17897$
Root an. cond. $1.50855$
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  − 2·3-s + 3·4-s + 2·5-s − 2·7-s + 3·9-s + 6·11-s − 6·12-s − 6·13-s − 4·15-s + 5·16-s − 8·17-s − 2·19-s + 6·20-s + 4·21-s + 8·23-s + 3·25-s − 4·27-s − 6·28-s + 2·29-s + 12·31-s − 12·33-s − 4·35-s + 9·36-s + 2·37-s + 12·39-s − 14·41-s + 6·43-s + ⋯
L(s)  = 1  − 1.15·3-s + 3/2·4-s + 0.894·5-s − 0.755·7-s + 9-s + 1.80·11-s − 1.73·12-s − 1.66·13-s − 1.03·15-s + 5/4·16-s − 1.94·17-s − 0.458·19-s + 1.34·20-s + 0.872·21-s + 1.66·23-s + 3/5·25-s − 0.769·27-s − 1.13·28-s + 0.371·29-s + 2.15·31-s − 2.08·33-s − 0.676·35-s + 3/2·36-s + 0.328·37-s + 1.92·39-s − 2.18·41-s + 0.914·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 81225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(81225\)    =    \(3^{2} \cdot 5^{2} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(5.17897\)
Root analytic conductor: \(1.50855\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 81225,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.623072106\)
\(L(\frac12)\) \(\approx\) \(1.623072106\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{2} \)
5$C_1$ \( ( 1 - T )^{2} \)
19$C_1$ \( ( 1 + T )^{2} \)
good2$C_2^2$ \( 1 - 3 T^{2} + p^{2} T^{4} \) 2.2.a_ad
7$D_{4}$ \( 1 + 2 T + 8 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.7.c_i
11$D_{4}$ \( 1 - 6 T + 24 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.11.ag_y
13$D_{4}$ \( 1 + 6 T + 28 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.13.g_bc
17$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.17.i_by
23$D_{4}$ \( 1 - 8 T + 34 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.23.ai_bi
29$D_{4}$ \( 1 - 2 T - 4 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.29.ac_ae
31$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.31.am_du
37$D_{4}$ \( 1 - 2 T + 68 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.37.ac_cq
41$D_{4}$ \( 1 + 14 T + 124 T^{2} + 14 p T^{3} + p^{2} T^{4} \) 2.41.o_eu
43$D_{4}$ \( 1 - 6 T + 88 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.43.ag_dk
47$D_{4}$ \( 1 - 8 T + 82 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.47.ai_de
53$D_{4}$ \( 1 - 12 T + 114 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.53.am_ek
59$D_{4}$ \( 1 - 12 T + 126 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.59.am_ew
61$D_{4}$ \( 1 + 12 T + 130 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.61.m_fa
67$D_{4}$ \( 1 - 8 T + 38 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.67.ai_bm
71$D_{4}$ \( 1 - 4 T + 118 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.71.ae_eo
73$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.73.au_jm
79$D_{4}$ \( 1 + 8 T + 62 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.79.i_ck
83$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.83.am_hu
89$D_{4}$ \( 1 + 18 T + 196 T^{2} + 18 p T^{3} + p^{2} T^{4} \) 2.89.s_ho
97$D_{4}$ \( 1 - 10 T + 156 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.97.ak_ga
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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

−12.13837975172374549503100235117, −11.38756509348376275999744915614, −11.36471851826579499463366400610, −10.74072057737776164736606977967, −10.18739812158716040263250652269, −9.949553341683311924501606218120, −9.308506745153719777501357755781, −8.957272504747976715941407857687, −8.252292213850488453803911291376, −7.07298766388026975824108747754, −7.02375917117060883853258051725, −6.59347835013690321556803685122, −6.39052012354167499822551865270, −5.79765047757978808166933689828, −4.91150139506397591927910874452, −4.61530551040323117217267653120, −3.65404537443948603368575444572, −2.54350021810836172772364651152, −2.26979476609353166891886826979, −1.09419492147748949812882672059, 1.09419492147748949812882672059, 2.26979476609353166891886826979, 2.54350021810836172772364651152, 3.65404537443948603368575444572, 4.61530551040323117217267653120, 4.91150139506397591927910874452, 5.79765047757978808166933689828, 6.39052012354167499822551865270, 6.59347835013690321556803685122, 7.02375917117060883853258051725, 7.07298766388026975824108747754, 8.252292213850488453803911291376, 8.957272504747976715941407857687, 9.308506745153719777501357755781, 9.949553341683311924501606218120, 10.18739812158716040263250652269, 10.74072057737776164736606977967, 11.36471851826579499463366400610, 11.38756509348376275999744915614, 12.13837975172374549503100235117

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