↑

Properties

Label 4-6480e2-1.1-c1e2-0-6
Degree $4$
Conductor $41990400$
Sign $1$
Analytic cond. $2677.34$
Root an. cond. $7.19326$
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  + 2·5-s − 7-s − 5·11-s + 5·13-s + 4·17-s − 2·19-s + 7·23-s + 3·25-s + 5·29-s − 5·31-s − 2·35-s − 6·37-s − 8·43-s − 5·47-s + 49-s + 9·53-s − 10·55-s + 26·59-s + 2·61-s + 10·65-s + 14·67-s + 9·71-s + 5·77-s − 6·79-s − 10·83-s + 8·85-s + 3·89-s + ⋯
L(s)  = 1  + 0.894·5-s − 0.377·7-s − 1.50·11-s + 1.38·13-s + 0.970·17-s − 0.458·19-s + 1.45·23-s + 3/5·25-s + 0.928·29-s − 0.898·31-s − 0.338·35-s − 0.986·37-s − 1.21·43-s − 0.729·47-s + 1/7·49-s + 1.23·53-s − 1.34·55-s + 3.38·59-s + 0.256·61-s + 1.24·65-s + 1.71·67-s + 1.06·71-s + 0.569·77-s − 0.675·79-s − 1.09·83-s + 0.867·85-s + 0.317·89-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 41990400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 41990400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(41990400\)    =    \(2^{8} \cdot 3^{8} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(2677.34\)
Root analytic conductor: \(7.19326\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 41990400,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.301156603\)
\(L(\frac12)\) \(\approx\) \(4.301156603\)
\(L(\frac{3}{2})\) 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 \( 1 \)
3 \( 1 \)
5$C_1$ \( ( 1 - T )^{2} \)
good7$D_{4}$ \( 1 + T + p T^{3} + p^{2} T^{4} \) 2.7.b_a
11$C_2^2$ \( 1 + 5 T + 14 T^{2} + 5 p T^{3} + p^{2} T^{4} \) 2.11.f_o
13$D_{4}$ \( 1 - 5 T + 18 T^{2} - 5 p T^{3} + p^{2} T^{4} \) 2.13.af_s
17$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.17.ae_bm
19$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.19.c_bn
23$D_{4}$ \( 1 - 7 T + 44 T^{2} - 7 p T^{3} + p^{2} T^{4} \) 2.23.ah_bs
29$D_{4}$ \( 1 - 5 T + 50 T^{2} - 5 p T^{3} + p^{2} T^{4} \) 2.29.af_by
31$D_{4}$ \( 1 + 5 T + 54 T^{2} + 5 p T^{3} + p^{2} T^{4} \) 2.31.f_cc
37$D_{4}$ \( 1 + 6 T + 26 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.37.g_ba
41$C_2^2$ \( 1 + 25 T^{2} + p^{2} T^{4} \) 2.41.a_z
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.43.i_dy
47$D_{4}$ \( 1 + 5 T + 86 T^{2} + 5 p T^{3} + p^{2} T^{4} \) 2.47.f_di
53$D_{4}$ \( 1 - 9 T + 112 T^{2} - 9 p T^{3} + p^{2} T^{4} \) 2.53.aj_ei
59$C_2$ \( ( 1 - 13 T + p T^{2} )^{2} \) 2.59.aba_lb
61$D_{4}$ \( 1 - 2 T + 66 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.61.ac_co
67$D_{4}$ \( 1 - 14 T + 126 T^{2} - 14 p T^{3} + p^{2} T^{4} \) 2.67.ao_ew
71$D_{4}$ \( 1 - 9 T + 148 T^{2} - 9 p T^{3} + p^{2} T^{4} \) 2.71.aj_fs
73$C_2^2$ \( 1 - 82 T^{2} + p^{2} T^{4} \) 2.73.a_ade
79$D_{4}$ \( 1 + 6 T + 110 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.79.g_eg
83$D_{4}$ \( 1 + 10 T + 134 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.83.k_fe
89$D_{4}$ \( 1 - 3 T + 52 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.89.ad_ca
97$C_2$ \( ( 1 - 16 T + p T^{2} )^{2} \) 2.97.abg_ri
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

−8.184983694435237956185362496476, −8.062853262855139860539198620803, −7.37048299558096207084207961422, −7.04128196177843053936631471924, −6.76425706247148901615670475613, −6.54768077981291665050237455404, −6.01106787266463468182907304697, −5.55661514803866260797446248003, −5.41308993567024169202671242386, −5.17732153873955214037879328520, −4.74637930565131130713541210571, −4.18487038051319654468145275660, −3.52527771959397135454840513649, −3.51578324947737316990869052143, −2.95712391446607568717122170974, −2.58646043757033333191840839437, −1.96063809179226636968203703998, −1.77463562754797208096955677276, −0.77915743572500989599453285921, −0.72300981464788957430510522089, 0.72300981464788957430510522089, 0.77915743572500989599453285921, 1.77463562754797208096955677276, 1.96063809179226636968203703998, 2.58646043757033333191840839437, 2.95712391446607568717122170974, 3.51578324947737316990869052143, 3.52527771959397135454840513649, 4.18487038051319654468145275660, 4.74637930565131130713541210571, 5.17732153873955214037879328520, 5.41308993567024169202671242386, 5.55661514803866260797446248003, 6.01106787266463468182907304697, 6.54768077981291665050237455404, 6.76425706247148901615670475613, 7.04128196177843053936631471924, 7.37048299558096207084207961422, 8.062853262855139860539198620803, 8.184983694435237956185362496476

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