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Properties

Label 4-4232e2-1.1-c1e2-0-0
Degree $4$
Conductor $17909824$
Sign $1$
Analytic cond. $1141.94$
Root an. cond. $5.81314$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 6·5-s − 4·7-s + 2·13-s − 12·19-s + 17·25-s + 6·29-s − 12·31-s − 24·35-s − 16·37-s + 6·41-s − 4·43-s + 8·47-s + 4·49-s + 2·53-s − 4·59-s − 2·61-s + 12·65-s − 8·67-s − 4·71-s + 6·73-s − 24·79-s − 9·81-s − 4·83-s + 2·89-s − 8·91-s − 72·95-s − 6·97-s + ⋯
L(s)  = 1  + 2.68·5-s − 1.51·7-s + 0.554·13-s − 2.75·19-s + 17/5·25-s + 1.11·29-s − 2.15·31-s − 4.05·35-s − 2.63·37-s + 0.937·41-s − 0.609·43-s + 1.16·47-s + 4/7·49-s + 0.274·53-s − 0.520·59-s − 0.256·61-s + 1.48·65-s − 0.977·67-s − 0.474·71-s + 0.702·73-s − 2.70·79-s − 81-s − 0.439·83-s + 0.211·89-s − 0.838·91-s − 7.38·95-s − 0.609·97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 17909824 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17909824 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(17909824\)    =    \(2^{6} \cdot 23^{4}\)
Sign: $1$
Analytic conductor: \(1141.94\)
Root analytic conductor: \(5.81314\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 17909824,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.558586218\)
\(L(\frac12)\) \(\approx\) \(1.558586218\)
\(L(\frac{3}{2})\) 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 \( 1 \)
23 \( 1 \)
good3$C_2^2$ \( 1 + p^{2} T^{4} \) 2.3.a_a
5$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.5.ag_t
7$D_{4}$ \( 1 + 4 T + 12 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.7.e_m
11$C_2^2$ \( 1 + 16 T^{2} + p^{2} T^{4} \) 2.11.a_q
13$D_{4}$ \( 1 - 2 T + 3 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.13.ac_d
17$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.17.a_k
19$D_{4}$ \( 1 + 12 T + 68 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.19.m_cq
29$D_{4}$ \( 1 - 6 T + 43 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.29.ag_br
31$D_{4}$ \( 1 + 12 T + 74 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.31.m_cw
37$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.37.q_fi
41$D_{4}$ \( 1 - 6 T + 67 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.41.ag_cp
43$D_{4}$ \( 1 + 4 T + 66 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.43.e_co
47$D_{4}$ \( 1 - 8 T + 104 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.47.ai_ea
53$D_{4}$ \( 1 - 2 T + 11 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.53.ac_l
59$D_{4}$ \( 1 + 4 T + 68 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.59.e_cq
61$D_{4}$ \( 1 + 2 T + 99 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.61.c_dv
67$D_{4}$ \( 1 + 8 T + 54 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.67.i_cc
71$D_{4}$ \( 1 + 4 T + 140 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.71.e_fk
73$D_{4}$ \( 1 - 6 T + 131 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.73.ag_fb
79$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.79.y_lq
83$D_{4}$ \( 1 + 4 T + 146 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.83.e_fq
89$D_{4}$ \( 1 - 2 T + 83 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.89.ac_df
97$D_{4}$ \( 1 + 6 T + 179 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.97.g_gx
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.702079898725639516547165003243, −8.644784361350816394908239660807, −7.86511704549893568251717891863, −7.21524738914806835531603232493, −6.96874039584520235718485119670, −6.65475207581466440039092097850, −6.20614447526914423603655282753, −6.12432279211170521066373284489, −5.66050882712339332762094922317, −5.59114044853761489439379217976, −4.94328545718213615652406832514, −4.46910830411047177040635568021, −3.99728684480772504932059002909, −3.54272467555133588531006860234, −3.01092778331184209641833939746, −2.64339306117457798416552096483, −2.07704930830632050310166771055, −1.80895644369064784936923534508, −1.44951433902820413650524524863, −0.31577642208192868426932196748, 0.31577642208192868426932196748, 1.44951433902820413650524524863, 1.80895644369064784936923534508, 2.07704930830632050310166771055, 2.64339306117457798416552096483, 3.01092778331184209641833939746, 3.54272467555133588531006860234, 3.99728684480772504932059002909, 4.46910830411047177040635568021, 4.94328545718213615652406832514, 5.59114044853761489439379217976, 5.66050882712339332762094922317, 6.12432279211170521066373284489, 6.20614447526914423603655282753, 6.65475207581466440039092097850, 6.96874039584520235718485119670, 7.21524738914806835531603232493, 7.86511704549893568251717891863, 8.644784361350816394908239660807, 8.702079898725639516547165003243

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