Properties

Label 4-1860e2-1.1-c1e2-0-1
Degree $4$
Conductor $3459600$
Sign $1$
Analytic cond. $220.587$
Root an. cond. $3.85385$
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  − 3-s − 5-s + 4·11-s + 3·13-s + 15-s + 4·17-s + 5·19-s − 4·23-s + 27-s − 12·29-s + 4·31-s − 4·33-s − 7·37-s − 3·39-s + 8·41-s + 43-s + 12·47-s + 7·49-s − 4·51-s + 4·53-s − 4·55-s − 5·57-s + 20·61-s − 3·65-s + 4·67-s + 4·69-s + 6·71-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.447·5-s + 1.20·11-s + 0.832·13-s + 0.258·15-s + 0.970·17-s + 1.14·19-s − 0.834·23-s + 0.192·27-s − 2.22·29-s + 0.718·31-s − 0.696·33-s − 1.15·37-s − 0.480·39-s + 1.24·41-s + 0.152·43-s + 1.75·47-s + 49-s − 0.560·51-s + 0.549·53-s − 0.539·55-s − 0.662·57-s + 2.56·61-s − 0.372·65-s + 0.488·67-s + 0.481·69-s + 0.712·71-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3459600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3459600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(3459600\)    =    \(2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 31^{2}\)
Sign: $1$
Analytic conductor: \(220.587\)
Root analytic conductor: \(3.85385\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 3459600,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.312843846\)
\(L(\frac12)\) \(\approx\) \(2.312843846\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T + T^{2} \)
5$C_2$ \( 1 + T + T^{2} \)
31$C_2$ \( 1 - 4 T + p T^{2} \)
good7$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.7.a_ah
11$C_2^2$ \( 1 - 4 T + 5 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.11.ae_f
13$C_2^2$ \( 1 - 3 T - 4 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.13.ad_ae
17$C_2^2$ \( 1 - 4 T - T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.17.ae_ab
19$C_2^2$ \( 1 - 5 T + 6 T^{2} - 5 p T^{3} + p^{2} T^{4} \) 2.19.af_g
23$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.23.e_by
29$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.29.m_dq
37$C_2^2$ \( 1 + 7 T + 12 T^{2} + 7 p T^{3} + p^{2} T^{4} \) 2.37.h_m
41$C_2^2$ \( 1 - 8 T + 23 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.41.ai_x
43$C_2^2$ \( 1 - T - 42 T^{2} - p T^{3} + p^{2} T^{4} \) 2.43.ab_abq
47$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.47.am_fa
53$C_2^2$ \( 1 - 4 T - 37 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.53.ae_abl
59$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.59.a_ach
61$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.61.au_io
67$C_2^2$ \( 1 - 4 T - 51 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.67.ae_abz
71$C_2^2$ \( 1 - 6 T - 35 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.71.ag_abj
73$C_2$ \( ( 1 - 17 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.73.ah_ay
79$C_2^2$ \( 1 - 12 T + 65 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.79.am_cn
83$C_2^2$ \( 1 - 2 T - 79 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.83.ac_adb
89$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.89.ai_hm
97$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \) 2.97.as_kp
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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

−9.458845294266984180729975357966, −8.968424243209705889904749610090, −8.777576987284006642367863605650, −8.255384381760572554457279568417, −7.66721601174035139881331849444, −7.60736199192867486158505449666, −7.02109606437611604914713503569, −6.74462600671270894468855301098, −6.10915920228932421483499688273, −5.82380071692728006988432533841, −5.49660240013772622882155092793, −5.13001051292100262002988774942, −4.44022214496352150296741997319, −3.83188088838056662237899193439, −3.69981546431692415802750847098, −3.41215057028045922248812204269, −2.39791983285820431754999980172, −1.97347303620719470969677829244, −1.02065579828521175394748948964, −0.75406109417466985950559733146, 0.75406109417466985950559733146, 1.02065579828521175394748948964, 1.97347303620719470969677829244, 2.39791983285820431754999980172, 3.41215057028045922248812204269, 3.69981546431692415802750847098, 3.83188088838056662237899193439, 4.44022214496352150296741997319, 5.13001051292100262002988774942, 5.49660240013772622882155092793, 5.82380071692728006988432533841, 6.10915920228932421483499688273, 6.74462600671270894468855301098, 7.02109606437611604914713503569, 7.60736199192867486158505449666, 7.66721601174035139881331849444, 8.255384381760572554457279568417, 8.777576987284006642367863605650, 8.968424243209705889904749610090, 9.458845294266984180729975357966

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