↑

Properties

Label 4-1040e2-1.1-c1e2-0-48
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

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s + 2·5-s + 2·9-s − 2·11-s − 6·13-s + 4·15-s + 10·17-s + 6·19-s + 10·23-s − 25-s + 6·27-s − 2·31-s − 4·33-s − 12·39-s + 2·41-s − 10·43-s + 4·45-s + 10·49-s + 20·51-s − 2·53-s − 4·55-s + 12·57-s − 6·59-s + 4·61-s − 12·65-s + 24·67-s + 20·69-s + ⋯
L(s)  = 1  + 1.15·3-s + 0.894·5-s + 2/3·9-s − 0.603·11-s − 1.66·13-s + 1.03·15-s + 2.42·17-s + 1.37·19-s + 2.08·23-s − 1/5·25-s + 1.15·27-s − 0.359·31-s − 0.696·33-s − 1.92·39-s + 0.312·41-s − 1.52·43-s + 0.596·45-s + 10/7·49-s + 2.80·51-s − 0.274·53-s − 0.539·55-s + 1.58·57-s − 0.781·59-s + 0.512·61-s − 1.48·65-s + 2.93·67-s + 2.40·69-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

\(L(1)\) \(\approx\) \(4.223550711\)
\(L(\frac12)\) \(\approx\) \(4.223550711\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - 2 T + p T^{2} \)
13$C_2$ \( 1 + 6 T + p T^{2} \)
good3$C_2^2$ \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.3.ac_c
7$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.7.a_ak
11$C_2^2$ \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.11.c_c
17$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.17.ak_by
19$C_2^2$ \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.19.ag_s
23$C_2^2$ \( 1 - 10 T + 50 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.23.ak_by
29$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.29.a_abq
31$C_2^2$ \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.31.c_c
37$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.37.a_ak
41$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.41.ac_c
43$C_2^2$ \( 1 + 10 T + 50 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.43.k_by
47$C_2^2$ \( 1 - 90 T^{2} + p^{2} T^{4} \) 2.47.a_adm
53$C_2^2$ \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.53.c_c
59$C_2^2$ \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.59.g_s
61$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.61.ae_ew
67$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \) 2.67.ay_ks
71$C_2^2$ \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.71.c_c
73$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.73.m_ha
79$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \) 2.79.a_bm
83$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \) 2.83.a_afa
89$C_2^2$ \( 1 + 14 T + 98 T^{2} + 14 p T^{3} + p^{2} T^{4} \) 2.89.o_du
97$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.97.e_hq
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.931644001295561684392439987426, −9.798065959332908424143752101233, −9.274975215191132695786118611164, −9.135648543957699639987450501662, −8.320761161940728512150031666602, −8.196501321738638313456107242732, −7.46591539597376790860074822749, −7.45525057823728268072971380718, −6.95398333378524764986215428897, −6.45736257955145480009130294983, −5.52635646497015312772675803753, −5.30835866201649191293625867287, −5.23787504807791467295471590629, −4.49227910694247045450022414743, −3.69696213373275198721793127664, −3.08901496899527110706778186399, −2.90527725196037039093687466847, −2.42994484230978397411939588113, −1.57560381046418962044430841509, −0.936743170887227486651180314097, 0.936743170887227486651180314097, 1.57560381046418962044430841509, 2.42994484230978397411939588113, 2.90527725196037039093687466847, 3.08901496899527110706778186399, 3.69696213373275198721793127664, 4.49227910694247045450022414743, 5.23787504807791467295471590629, 5.30835866201649191293625867287, 5.52635646497015312772675803753, 6.45736257955145480009130294983, 6.95398333378524764986215428897, 7.45525057823728268072971380718, 7.46591539597376790860074822749, 8.196501321738638313456107242732, 8.320761161940728512150031666602, 9.135648543957699639987450501662, 9.274975215191132695786118611164, 9.798065959332908424143752101233, 9.931644001295561684392439987426

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