↑

Properties

Label 4-2240e2-1.1-c1e2-0-5
Degree $4$
Conductor $5017600$
Sign $1$
Analytic cond. $319.926$
Root an. cond. $4.22924$
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  + 3-s − 2·5-s + 2·7-s − 9-s + 11-s − 3·13-s − 2·15-s + 5·17-s + 2·21-s + 3·25-s − 29-s − 2·31-s + 33-s − 4·35-s − 6·37-s − 3·39-s + 4·41-s + 8·43-s + 2·45-s + 11·47-s + 3·49-s + 5·51-s − 18·53-s − 2·55-s − 4·59-s + 4·61-s − 2·63-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.894·5-s + 0.755·7-s − 1/3·9-s + 0.301·11-s − 0.832·13-s − 0.516·15-s + 1.21·17-s + 0.436·21-s + 3/5·25-s − 0.185·29-s − 0.359·31-s + 0.174·33-s − 0.676·35-s − 0.986·37-s − 0.480·39-s + 0.624·41-s + 1.21·43-s + 0.298·45-s + 1.60·47-s + 3/7·49-s + 0.700·51-s − 2.47·53-s − 0.269·55-s − 0.520·59-s + 0.512·61-s − 0.251·63-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5017600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5017600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(5017600\)    =    \(2^{12} \cdot 5^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(319.926\)
Root analytic conductor: \(4.22924\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 5017600,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.546937684\)
\(L(\frac12)\) \(\approx\) \(2.546937684\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{2} \)
7$C_1$ \( ( 1 - T )^{2} \)
good3$D_{4}$ \( 1 - T + 2 T^{2} - p T^{3} + p^{2} T^{4} \) 2.3.ab_c
11$D_{4}$ \( 1 - T + 18 T^{2} - p T^{3} + p^{2} T^{4} \) 2.11.ab_s
13$D_{4}$ \( 1 + 3 T + 24 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.13.d_y
17$D_{4}$ \( 1 - 5 T + 36 T^{2} - 5 p T^{3} + p^{2} T^{4} \) 2.17.af_bk
19$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.19.a_bm
23$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.23.a_bu
29$D_{4}$ \( 1 + T + 20 T^{2} + p T^{3} + p^{2} T^{4} \) 2.29.b_u
31$D_{4}$ \( 1 + 2 T + 46 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.31.c_bu
37$D_{4}$ \( 1 + 6 T + 66 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.37.g_co
41$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.41.ae_di
43$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.43.ai_dy
47$D_{4}$ \( 1 - 11 T + 86 T^{2} - 11 p T^{3} + p^{2} T^{4} \) 2.47.al_di
53$D_{4}$ \( 1 + 18 T + 170 T^{2} + 18 p T^{3} + p^{2} T^{4} \) 2.53.s_go
59$D_{4}$ \( 1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.59.e_cc
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.ae_cs
71$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.71.ai_gc
73$D_{4}$ \( 1 - 16 T + 142 T^{2} - 16 p T^{3} + p^{2} T^{4} \) 2.73.aq_fm
79$D_{4}$ \( 1 - 5 T + 58 T^{2} - 5 p T^{3} + p^{2} T^{4} \) 2.79.af_cg
83$D_{4}$ \( 1 - 4 T + 102 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.83.ae_dy
89$C_4$ \( 1 - 8 T + 126 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.89.ai_ew
97$D_{4}$ \( 1 + 3 T - 12 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.97.d_am
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

−9.049821303742649391373595156912, −8.983499220343970140623655125520, −8.271884798887795535386421100149, −8.096262004580016505674474027318, −7.68136757464873641979608708777, −7.53460299832476471237283423719, −7.02626111768266080919543392751, −6.67264519206290839108154027445, −5.93958644620165190051237719827, −5.77403380216881752251364300192, −5.10118796806901444798421924921, −4.83564056033958142797337169179, −4.42531352134838379430142830981, −3.85311041057147623817172685885, −3.34101825613025253251743713930, −3.23643208297523430129350864074, −2.29563371043980537327989334727, −2.15584734182179304000462146284, −1.20204463389652217505244716624, −0.58526933706795397978733388340, 0.58526933706795397978733388340, 1.20204463389652217505244716624, 2.15584734182179304000462146284, 2.29563371043980537327989334727, 3.23643208297523430129350864074, 3.34101825613025253251743713930, 3.85311041057147623817172685885, 4.42531352134838379430142830981, 4.83564056033958142797337169179, 5.10118796806901444798421924921, 5.77403380216881752251364300192, 5.93958644620165190051237719827, 6.67264519206290839108154027445, 7.02626111768266080919543392751, 7.53460299832476471237283423719, 7.68136757464873641979608708777, 8.096262004580016505674474027318, 8.271884798887795535386421100149, 8.983499220343970140623655125520, 9.049821303742649391373595156912

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