↑

Properties

Label 4-33e4-1.1-c1e2-0-10
Degree $4$
Conductor $1185921$
Sign $1$
Analytic cond. $75.6153$
Root an. cond. $2.94884$
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  + 3·2-s + 4·4-s − 5-s − 2·7-s + 3·8-s − 3·10-s − 4·13-s − 6·14-s + 3·16-s + 9·17-s + 5·19-s − 4·20-s + 4·23-s − 8·25-s − 12·26-s − 8·28-s + 12·29-s − 31-s + 6·32-s + 27·34-s + 2·35-s − 8·37-s + 15·38-s − 3·40-s + 4·41-s + 12·46-s + 9·47-s + ⋯
L(s)  = 1  + 2.12·2-s + 2·4-s − 0.447·5-s − 0.755·7-s + 1.06·8-s − 0.948·10-s − 1.10·13-s − 1.60·14-s + 3/4·16-s + 2.18·17-s + 1.14·19-s − 0.894·20-s + 0.834·23-s − 8/5·25-s − 2.35·26-s − 1.51·28-s + 2.22·29-s − 0.179·31-s + 1.06·32-s + 4.63·34-s + 0.338·35-s − 1.31·37-s + 2.43·38-s − 0.474·40-s + 0.624·41-s + 1.76·46-s + 1.31·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1185921 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1185921 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1185921\)    =    \(3^{4} \cdot 11^{4}\)
Sign: $1$
Analytic conductor: \(75.6153\)
Root analytic conductor: \(2.94884\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1185921,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(5.443253550\)
\(L(\frac12)\) \(\approx\) \(5.443253550\)
\(L(\frac{3}{2})\) 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 $\mathbf{F}_p$
bad3 \( 1 \)
11 \( 1 \)
good2$C_2^2$ \( 1 - 3 T + 5 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.2.ad_f
5$D_{4}$ \( 1 + T + 9 T^{2} + p T^{3} + p^{2} T^{4} \) 2.5.b_j
7$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.7.c_p
13$D_{4}$ \( 1 + 4 T + 25 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.13.e_z
17$D_{4}$ \( 1 - 9 T + 43 T^{2} - 9 p T^{3} + p^{2} T^{4} \) 2.17.aj_br
19$D_{4}$ \( 1 - 5 T + 33 T^{2} - 5 p T^{3} + p^{2} T^{4} \) 2.19.af_bh
23$D_{4}$ \( 1 - 4 T + 45 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.23.ae_bt
29$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.29.am_dq
31$D_{4}$ \( 1 + T + p T^{2} + p T^{3} + p^{2} T^{4} \) 2.31.b_bf
37$D_{4}$ \( 1 + 8 T + 85 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.37.i_dh
41$D_{4}$ \( 1 - 4 T + 81 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.41.ae_dd
43$C_2^2$ \( 1 + 41 T^{2} + p^{2} T^{4} \) 2.43.a_bp
47$D_{4}$ \( 1 - 9 T + 83 T^{2} - 9 p T^{3} + p^{2} T^{4} \) 2.47.aj_df
53$D_{4}$ \( 1 - 3 T + 107 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.53.ad_ed
59$D_{4}$ \( 1 + 17 T + 189 T^{2} + 17 p T^{3} + p^{2} T^{4} \) 2.59.r_hh
61$D_{4}$ \( 1 + 3 T + 23 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.61.d_x
67$D_{4}$ \( 1 + 3 T + 125 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.67.d_ev
71$D_{4}$ \( 1 + 5 T + 87 T^{2} + 5 p T^{3} + p^{2} T^{4} \) 2.71.f_dj
73$D_{4}$ \( 1 - 2 T + 102 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.73.ac_dy
79$C_2$ \( ( 1 - 11 T + p T^{2} )^{2} \) 2.79.aw_kt
83$D_{4}$ \( 1 - 6 T + 155 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.83.ag_fz
89$D_{4}$ \( 1 - 12 T + 209 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.89.am_ib
97$D_{4}$ \( 1 - 9 T + 203 T^{2} - 9 p T^{3} + p^{2} T^{4} \) 2.97.aj_hv
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

−10.14857687323640458070806657472, −9.810206682881451836354678752098, −9.168084567698295555309271447297, −9.154018537885521861521306796922, −8.060283860149454633022001357399, −7.79984322600123080268866465916, −7.62291945162226335947574846943, −7.05773190493427797799574538686, −6.43985475898287988206672321166, −6.17509592723152359241586316877, −5.50854143133989056738374764122, −5.36748558817282313995227844588, −4.71508052510387465339825702949, −4.66588366010918778643755879386, −3.68373839018247499821777243935, −3.63017055291950575811495136813, −2.96283532031148272585072372559, −2.84625615125432410153973702590, −1.70620894248544252523116471375, −0.76025839619993834923995410622, 0.76025839619993834923995410622, 1.70620894248544252523116471375, 2.84625615125432410153973702590, 2.96283532031148272585072372559, 3.63017055291950575811495136813, 3.68373839018247499821777243935, 4.66588366010918778643755879386, 4.71508052510387465339825702949, 5.36748558817282313995227844588, 5.50854143133989056738374764122, 6.17509592723152359241586316877, 6.43985475898287988206672321166, 7.05773190493427797799574538686, 7.62291945162226335947574846943, 7.79984322600123080268866465916, 8.060283860149454633022001357399, 9.154018537885521861521306796922, 9.168084567698295555309271447297, 9.810206682881451836354678752098, 10.14857687323640458070806657472

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