↑

Properties

Label 4-2299e2-1.1-c1e2-0-2
Degree $4$
Conductor $5285401$
Sign $1$
Analytic cond. $337.001$
Root an. cond. $4.28457$
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 + 4-s − 5-s − 3·7-s + 2·9-s + 2·12-s + 4·13-s − 2·15-s − 3·16-s + 5·17-s + 2·19-s − 20-s − 6·21-s + 23-s − 8·25-s + 6·27-s − 3·28-s + 8·29-s + 16·31-s + 3·35-s + 2·36-s − 4·37-s + 8·39-s + 10·41-s + 5·43-s − 2·45-s + 15·47-s + ⋯
L(s)  = 1  + 1.15·3-s + 1/2·4-s − 0.447·5-s − 1.13·7-s + 2/3·9-s + 0.577·12-s + 1.10·13-s − 0.516·15-s − 3/4·16-s + 1.21·17-s + 0.458·19-s − 0.223·20-s − 1.30·21-s + 0.208·23-s − 8/5·25-s + 1.15·27-s − 0.566·28-s + 1.48·29-s + 2.87·31-s + 0.507·35-s + 1/3·36-s − 0.657·37-s + 1.28·39-s + 1.56·41-s + 0.762·43-s − 0.298·45-s + 2.18·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5285401 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5285401 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(5285401\)    =    \(11^{4} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(337.001\)
Root analytic conductor: \(4.28457\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 5285401,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.769159153\)
\(L(\frac12)\) \(\approx\) \(4.769159153\)
\(L(\frac{3}{2})\) 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 $\mathbf{F}_p$
bad11 \( 1 \)
19$C_1$ \( ( 1 - T )^{2} \)
good2$C_2^2$ \( 1 - T^{2} + p^{2} T^{4} \) 2.2.a_ab
3$C_2^2$ \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.3.ac_c
5$D_{4}$ \( 1 + T + 9 T^{2} + p T^{3} + p^{2} T^{4} \) 2.5.b_j
7$D_{4}$ \( 1 + 3 T + 15 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.7.d_p
13$D_{4}$ \( 1 - 4 T + 10 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.13.ae_k
17$D_{4}$ \( 1 - 5 T + 9 T^{2} - 5 p T^{3} + p^{2} T^{4} \) 2.17.af_j
23$D_{4}$ \( 1 - T + 45 T^{2} - p T^{3} + p^{2} T^{4} \) 2.23.ab_bt
29$C_4$ \( 1 - 8 T + 54 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.29.ai_cc
31$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.31.aq_ew
37$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.37.e_da
41$D_{4}$ \( 1 - 10 T + 62 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.41.ak_ck
43$D_{4}$ \( 1 - 5 T + 61 T^{2} - 5 p T^{3} + p^{2} T^{4} \) 2.43.af_cj
47$D_{4}$ \( 1 - 15 T + 149 T^{2} - 15 p T^{3} + p^{2} T^{4} \) 2.47.ap_ft
53$D_{4}$ \( 1 - 12 T + 122 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.53.am_es
59$D_{4}$ \( 1 - 16 T + 162 T^{2} - 16 p T^{3} + p^{2} T^{4} \) 2.59.aq_gg
61$D_{4}$ \( 1 - 15 T + 177 T^{2} - 15 p T^{3} + p^{2} T^{4} \) 2.61.ap_gv
67$D_{4}$ \( 1 + 8 T + 130 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.67.i_fa
71$D_{4}$ \( 1 - 10 T + 122 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.71.ak_es
73$D_{4}$ \( 1 - 16 T + 190 T^{2} - 16 p T^{3} + p^{2} T^{4} \) 2.73.aq_hi
79$D_{4}$ \( 1 + 12 T + 114 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.79.m_ek
83$D_{4}$ \( 1 + 5 T + 111 T^{2} + 5 p T^{3} + p^{2} T^{4} \) 2.83.f_eh
89$D_{4}$ \( 1 + 2 T + 174 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.89.c_gs
97$D_{4}$ \( 1 + 2 T + 70 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.97.c_cs
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

−9.040363479089810858807368543944, −8.822636460965089713695242898439, −8.340974137910220583871440336367, −8.121702973811623717606672642092, −7.84001890070502988200274477193, −7.19856762231216442533673179496, −6.96602839200597526463022295762, −6.54455841481763229551518168129, −6.22000145760170726518582657175, −5.75474493766509758148303136094, −5.35112146065011846909959060742, −4.66344281601966293857878682370, −4.06320088520080266305743706101, −3.91372492760317709257303183071, −3.43684680778838777387181267273, −2.75634239366939067647619850515, −2.69020502529472296104486305512, −2.21891341212330389881719538831, −1.01561239181170106658328356964, −0.893798454361588326556054343906, 0.893798454361588326556054343906, 1.01561239181170106658328356964, 2.21891341212330389881719538831, 2.69020502529472296104486305512, 2.75634239366939067647619850515, 3.43684680778838777387181267273, 3.91372492760317709257303183071, 4.06320088520080266305743706101, 4.66344281601966293857878682370, 5.35112146065011846909959060742, 5.75474493766509758148303136094, 6.22000145760170726518582657175, 6.54455841481763229551518168129, 6.96602839200597526463022295762, 7.19856762231216442533673179496, 7.84001890070502988200274477193, 8.121702973811623717606672642092, 8.340974137910220583871440336367, 8.822636460965089713695242898439, 9.040363479089810858807368543944

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