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Properties

Label 4-4640e2-1.1-c1e2-0-0
Degree $4$
Conductor $21529600$
Sign $1$
Analytic cond. $1372.74$
Root an. cond. $6.08691$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 2·5-s + 7-s − 4·9-s − 2·11-s + 13-s + 2·15-s − 5·17-s + 10·19-s − 21-s + 5·23-s + 3·25-s + 6·27-s + 2·29-s − 31-s + 2·33-s − 2·35-s + 14·37-s − 39-s + 6·41-s − 13·43-s + 8·45-s − 16·47-s − 2·49-s + 5·51-s + 3·53-s + 4·55-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.894·5-s + 0.377·7-s − 4/3·9-s − 0.603·11-s + 0.277·13-s + 0.516·15-s − 1.21·17-s + 2.29·19-s − 0.218·21-s + 1.04·23-s + 3/5·25-s + 1.15·27-s + 0.371·29-s − 0.179·31-s + 0.348·33-s − 0.338·35-s + 2.30·37-s − 0.160·39-s + 0.937·41-s − 1.98·43-s + 1.19·45-s − 2.33·47-s − 2/7·49-s + 0.700·51-s + 0.412·53-s + 0.539·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 21529600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21529600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(21529600\)    =    \(2^{10} \cdot 5^{2} \cdot 29^{2}\)
Sign: $1$
Analytic conductor: \(1372.74\)
Root analytic conductor: \(6.08691\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 21529600,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.257702516\)
\(L(\frac12)\) \(\approx\) \(1.257702516\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5$C_1$ \( ( 1 + T )^{2} \)
29$C_1$ \( ( 1 - T )^{2} \)
good3$D_{4}$ \( 1 + T + 5 T^{2} + p T^{3} + p^{2} T^{4} \) 2.3.b_f
7$D_{4}$ \( 1 - T + 3 T^{2} - p T^{3} + p^{2} T^{4} \) 2.7.ab_d
11$D_{4}$ \( 1 + 2 T + 18 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.11.c_s
13$D_{4}$ \( 1 - T - 5 T^{2} - p T^{3} + p^{2} T^{4} \) 2.13.ab_af
17$D_{4}$ \( 1 + 5 T + 39 T^{2} + 5 p T^{3} + p^{2} T^{4} \) 2.17.f_bn
19$D_{4}$ \( 1 - 10 T + 58 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.19.ak_cg
23$D_{4}$ \( 1 - 5 T + 41 T^{2} - 5 p T^{3} + p^{2} T^{4} \) 2.23.af_bp
31$D_{4}$ \( 1 + T + 61 T^{2} + p T^{3} + p^{2} T^{4} \) 2.31.b_cj
37$D_{4}$ \( 1 - 14 T + 118 T^{2} - 14 p T^{3} + p^{2} T^{4} \) 2.37.ao_eo
41$D_{4}$ \( 1 - 6 T + 46 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.41.ag_bu
43$D_{4}$ \( 1 + 13 T + 127 T^{2} + 13 p T^{3} + p^{2} T^{4} \) 2.43.n_ex
47$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.47.q_gc
53$D_{4}$ \( 1 - 3 T + 47 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.53.ad_bv
59$D_{4}$ \( 1 + 21 T + 227 T^{2} + 21 p T^{3} + p^{2} T^{4} \) 2.59.v_it
61$D_{4}$ \( 1 - 11 T + 141 T^{2} - 11 p T^{3} + p^{2} T^{4} \) 2.61.al_fl
67$D_{4}$ \( 1 - 4 T + 118 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.67.ae_eo
71$D_{4}$ \( 1 - 8 T + 78 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.71.ai_da
73$D_{4}$ \( 1 - 15 T + 171 T^{2} - 15 p T^{3} + p^{2} T^{4} \) 2.73.ap_gp
79$C_4$ \( 1 + 17 T + 199 T^{2} + 17 p T^{3} + p^{2} T^{4} \) 2.79.r_hr
83$D_{4}$ \( 1 - 18 T + 202 T^{2} - 18 p T^{3} + p^{2} T^{4} \) 2.83.as_hu
89$D_{4}$ \( 1 - 10 T + 158 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.89.ak_gc
97$D_{4}$ \( 1 + 7 T + 55 T^{2} + 7 p T^{3} + p^{2} T^{4} \) 2.97.h_cd
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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

−8.257909003371320681937299431989, −8.173216749473993089222220521414, −7.76089954674312542278536624041, −7.61899310432646710402783165390, −6.94537000473268447033706961348, −6.72764991520718425392969516766, −6.23394395621112211322025120020, −6.06506034261886190020490069130, −5.35767985727673718589448552069, −5.21350721456265352209923151200, −4.74865931687511623892018378853, −4.72374037737587635787593775721, −3.93479475715607509213061223858, −3.48608530734305423702668303049, −2.99219611811788006512549636518, −2.92237715625787114604272229517, −2.27875871819941325928168641245, −1.55845095112198260314201711669, −0.857840336034160469912906834871, −0.42649675330621668308101257907, 0.42649675330621668308101257907, 0.857840336034160469912906834871, 1.55845095112198260314201711669, 2.27875871819941325928168641245, 2.92237715625787114604272229517, 2.99219611811788006512549636518, 3.48608530734305423702668303049, 3.93479475715607509213061223858, 4.72374037737587635787593775721, 4.74865931687511623892018378853, 5.21350721456265352209923151200, 5.35767985727673718589448552069, 6.06506034261886190020490069130, 6.23394395621112211322025120020, 6.72764991520718425392969516766, 6.94537000473268447033706961348, 7.61899310432646710402783165390, 7.76089954674312542278536624041, 8.173216749473993089222220521414, 8.257909003371320681937299431989

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