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Properties

Label 4-66e4-1.1-c1e2-0-3
Degree $4$
Conductor $18974736$
Sign $1$
Analytic cond. $1209.84$
Root an. cond. $5.89769$
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  + 5-s + 7-s − 7·13-s + 7·17-s − 19-s − 4·23-s − 8·25-s + 15·29-s − 5·31-s + 35-s + 3·37-s + 15·41-s − 5·47-s − 2·49-s − 3·53-s + 9·59-s − 7·61-s − 7·65-s + 8·67-s + 15·71-s + 13·73-s + 21·79-s − 3·83-s + 7·85-s + 8·89-s − 7·91-s − 95-s + ⋯
L(s)  = 1  + 0.447·5-s + 0.377·7-s − 1.94·13-s + 1.69·17-s − 0.229·19-s − 0.834·23-s − 8/5·25-s + 2.78·29-s − 0.898·31-s + 0.169·35-s + 0.493·37-s + 2.34·41-s − 0.729·47-s − 2/7·49-s − 0.412·53-s + 1.17·59-s − 0.896·61-s − 0.868·65-s + 0.977·67-s + 1.78·71-s + 1.52·73-s + 2.36·79-s − 0.329·83-s + 0.759·85-s + 0.847·89-s − 0.733·91-s − 0.102·95-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 18974736 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18974736 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(18974736\)    =    \(2^{4} \cdot 3^{4} \cdot 11^{4}\)
Sign: $1$
Analytic conductor: \(1209.84\)
Root analytic conductor: \(5.89769\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 18974736,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.076515719\)
\(L(\frac12)\) \(\approx\) \(3.076515719\)
\(L(\frac{3}{2})\) 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 \( 1 \)
11 \( 1 \)
good5$D_{4}$ \( 1 - T + 9 T^{2} - p T^{3} + p^{2} T^{4} \) 2.5.ab_j
7$D_{4}$ \( 1 - T + 3 T^{2} - p T^{3} + p^{2} T^{4} \) 2.7.ab_d
13$D_{4}$ \( 1 + 7 T + 37 T^{2} + 7 p T^{3} + p^{2} T^{4} \) 2.13.h_bl
17$D_{4}$ \( 1 - 7 T + 45 T^{2} - 7 p T^{3} + p^{2} T^{4} \) 2.17.ah_bt
19$D_{4}$ \( 1 + T + 27 T^{2} + p T^{3} + p^{2} T^{4} \) 2.19.b_bb
23$D_{4}$ \( 1 + 4 T + 30 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.23.e_be
29$D_{4}$ \( 1 - 15 T + 113 T^{2} - 15 p T^{3} + p^{2} T^{4} \) 2.29.ap_ej
31$D_{4}$ \( 1 + 5 T + 57 T^{2} + 5 p T^{3} + p^{2} T^{4} \) 2.31.f_cf
37$D_{4}$ \( 1 - 3 T + 65 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.37.ad_cn
41$D_{4}$ \( 1 - 15 T + 137 T^{2} - 15 p T^{3} + p^{2} T^{4} \) 2.41.ap_fh
43$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.43.a_di
47$D_{4}$ \( 1 + 5 T + 99 T^{2} + 5 p T^{3} + p^{2} T^{4} \) 2.47.f_dv
53$D_{4}$ \( 1 + 3 T + 77 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.53.d_cz
59$D_{4}$ \( 1 - 9 T + 107 T^{2} - 9 p T^{3} + p^{2} T^{4} \) 2.59.aj_ed
61$D_{4}$ \( 1 + 7 T + 133 T^{2} + 7 p T^{3} + p^{2} T^{4} \) 2.61.h_fd
67$D_{4}$ \( 1 - 8 T + 70 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.67.ai_cs
71$D_{4}$ \( 1 - 15 T + 197 T^{2} - 15 p T^{3} + p^{2} T^{4} \) 2.71.ap_hp
73$D_{4}$ \( 1 - 13 T + 157 T^{2} - 13 p T^{3} + p^{2} T^{4} \) 2.73.an_gb
79$D_{4}$ \( 1 - 21 T + 257 T^{2} - 21 p T^{3} + p^{2} T^{4} \) 2.79.av_jx
83$D_{4}$ \( 1 + 3 T - 43 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.83.d_abr
89$D_{4}$ \( 1 - 8 T + 174 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.89.ai_gs
97$D_{4}$ \( 1 + 9 T + 113 T^{2} + 9 p T^{3} + p^{2} T^{4} \) 2.97.j_ej
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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

−8.230296435743913011258667223485, −8.126099064893724119841317941485, −7.83271931270165027779312047305, −7.71351445350967229933942099868, −6.97089513123132957507482796486, −6.90162377916937381849479000719, −6.14656007453774871593201500837, −6.14553267688920410806584774095, −5.51744444602093447660257319843, −5.27063672756006939258765107135, −4.80332987721248054663746191499, −4.59515418806816908381230011322, −3.95089446364963540412954308466, −3.69112987199945919606313725728, −3.01697663058927967943507757325, −2.65348496370682255698238907470, −2.05714554698783872521527429353, −1.98538910301970712839937910170, −0.981837817849643709344626031661, −0.57274319524579901950640306173, 0.57274319524579901950640306173, 0.981837817849643709344626031661, 1.98538910301970712839937910170, 2.05714554698783872521527429353, 2.65348496370682255698238907470, 3.01697663058927967943507757325, 3.69112987199945919606313725728, 3.95089446364963540412954308466, 4.59515418806816908381230011322, 4.80332987721248054663746191499, 5.27063672756006939258765107135, 5.51744444602093447660257319843, 6.14553267688920410806584774095, 6.14656007453774871593201500837, 6.90162377916937381849479000719, 6.97089513123132957507482796486, 7.71351445350967229933942099868, 7.83271931270165027779312047305, 8.126099064893724119841317941485, 8.230296435743913011258667223485

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