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Properties

Label 4-1040e2-1.1-c1e2-0-25
Degree $4$
Conductor $1081600$
Sign $1$
Analytic cond. $68.9637$
Root an. cond. $2.88174$
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 − 3·7-s + 3·9-s − 3·11-s − 7·13-s + 4·17-s + 7·19-s − 4·23-s + 3·25-s + 8·29-s + 20·31-s − 6·35-s − 3·37-s + 2·41-s + 6·43-s + 6·45-s + 2·47-s + 7·49-s − 18·53-s − 6·55-s − 4·59-s + 14·61-s − 9·63-s − 14·65-s + 4·67-s + 6·71-s + 8·73-s + ⋯
L(s)  = 1  + 0.894·5-s − 1.13·7-s + 9-s − 0.904·11-s − 1.94·13-s + 0.970·17-s + 1.60·19-s − 0.834·23-s + 3/5·25-s + 1.48·29-s + 3.59·31-s − 1.01·35-s − 0.493·37-s + 0.312·41-s + 0.914·43-s + 0.894·45-s + 0.291·47-s + 49-s − 2.47·53-s − 0.809·55-s − 0.520·59-s + 1.79·61-s − 1.13·63-s − 1.73·65-s + 0.488·67-s + 0.712·71-s + 0.936·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1081600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1081600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1081600\)    =    \(2^{8} \cdot 5^{2} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(68.9637\)
Root analytic conductor: \(2.88174\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1081600,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.183355377\)
\(L(\frac12)\) \(\approx\) \(2.183355377\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{2} \)
13$C_2$ \( 1 + 7 T + p T^{2} \)
good3$C_2$ \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) 2.3.a_ad
7$C_2^2$ \( 1 + 3 T + 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.7.d_c
11$C_2^2$ \( 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.11.d_ac
17$C_2^2$ \( 1 - 4 T - T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.17.ae_ab
19$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + T + p T^{2} ) \) 2.19.ah_be
23$C_2^2$ \( 1 + 4 T - 7 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.23.e_ah
29$C_2^2$ \( 1 - 8 T + 35 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.29.ai_bj
31$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.31.au_gg
37$C_2^2$ \( 1 + 3 T - 28 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.37.d_abc
41$C_2^2$ \( 1 - 2 T - 37 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.41.ac_abl
43$C_2^2$ \( 1 - 6 T - 7 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.43.ag_ah
47$C_2$ \( ( 1 - T + p T^{2} )^{2} \) 2.47.ac_dr
53$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \) 2.53.s_hf
59$C_2^2$ \( 1 + 4 T - 43 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.59.e_abr
61$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 - T + p T^{2} ) \) 2.61.ao_ff
67$C_2^2$ \( 1 - 4 T - 51 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.67.ae_abz
71$C_2^2$ \( 1 - 6 T - 35 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.71.ag_abj
73$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.73.ai_gg
79$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.79.u_jy
83$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.83.a_gk
89$C_2^2$ \( 1 + T - 88 T^{2} + p T^{3} + p^{2} T^{4} \) 2.89.b_adk
97$C_2^2$ \( 1 + 16 T + 159 T^{2} + 16 p T^{3} + p^{2} T^{4} \) 2.97.q_gd
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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

−9.900096912838660520424409182755, −9.828803287047650627160180626082, −9.633045943457412736749359149404, −9.174127321069994995867871803631, −8.264873616973091332588064471588, −8.067446308438749830899649893147, −7.67134612970128390849417748114, −7.10228348192114430904729985552, −6.77431777862100081282101146224, −6.43425235774601177400295620642, −5.79759526651364542510291869715, −5.47325668756707026351300984270, −4.76110135074437241919592529354, −4.74128519240617665761652416027, −3.96381096315618940633706225952, −3.11832468131058604268564139473, −2.62234119848420678874917353043, −2.61954838010685489440722603534, −1.43219173246558327857857503143, −0.70755527216395934544313643734, 0.70755527216395934544313643734, 1.43219173246558327857857503143, 2.61954838010685489440722603534, 2.62234119848420678874917353043, 3.11832468131058604268564139473, 3.96381096315618940633706225952, 4.74128519240617665761652416027, 4.76110135074437241919592529354, 5.47325668756707026351300984270, 5.79759526651364542510291869715, 6.43425235774601177400295620642, 6.77431777862100081282101146224, 7.10228348192114430904729985552, 7.67134612970128390849417748114, 8.067446308438749830899649893147, 8.264873616973091332588064471588, 9.174127321069994995867871803631, 9.633045943457412736749359149404, 9.828803287047650627160180626082, 9.900096912838660520424409182755

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