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Properties

Label 4-8379e2-1.1-c1e2-0-5
Degree $4$
Conductor $70207641$
Sign $1$
Analytic cond. $4476.50$
Root an. cond. $8.17964$
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  + 2·2-s + 4-s + 2·5-s + 4·10-s − 2·11-s − 4·13-s + 16-s − 8·17-s + 2·19-s + 2·20-s − 4·22-s + 14·23-s − 7·25-s − 8·26-s + 16·29-s + 4·31-s − 2·32-s − 16·34-s + 4·37-s + 4·38-s − 12·41-s + 2·43-s − 2·44-s + 28·46-s − 6·47-s − 14·50-s − 4·52-s + ⋯
L(s)  = 1  + 1.41·2-s + 1/2·4-s + 0.894·5-s + 1.26·10-s − 0.603·11-s − 1.10·13-s + 1/4·16-s − 1.94·17-s + 0.458·19-s + 0.447·20-s − 0.852·22-s + 2.91·23-s − 7/5·25-s − 1.56·26-s + 2.97·29-s + 0.718·31-s − 0.353·32-s − 2.74·34-s + 0.657·37-s + 0.648·38-s − 1.87·41-s + 0.304·43-s − 0.301·44-s + 4.12·46-s − 0.875·47-s − 1.97·50-s − 0.554·52-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 70207641 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 70207641 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(70207641\)    =    \(3^{4} \cdot 7^{4} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(4476.50\)
Root analytic conductor: \(8.17964\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 70207641,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(5.653337906\)
\(L(\frac12)\) \(\approx\) \(5.653337906\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 \)
19$C_1$ \( ( 1 - T )^{2} \)
good2$D_{4}$ \( 1 - p T + 3 T^{2} - p^{2} T^{3} + p^{2} T^{4} \) 2.2.ac_d
5$C_2$ \( ( 1 - T + p T^{2} )^{2} \) 2.5.ac_l
11$D_{4}$ \( 1 + 2 T + 21 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.11.c_v
13$D_{4}$ \( 1 + 4 T + 12 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.13.e_m
17$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.17.i_by
23$D_{4}$ \( 1 - 14 T + 93 T^{2} - 14 p T^{3} + p^{2} T^{4} \) 2.23.ao_dp
29$D_{4}$ \( 1 - 16 T + 120 T^{2} - 16 p T^{3} + p^{2} T^{4} \) 2.29.aq_eq
31$D_{4}$ \( 1 - 4 T + 48 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.31.ae_bw
37$D_{4}$ \( 1 - 4 T + 28 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.37.ae_bc
41$D_{4}$ \( 1 + 12 T + 110 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.41.m_eg
43$D_{4}$ \( 1 - 2 T + 37 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.43.ac_bl
47$D_{4}$ \( 1 + 6 T + 101 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.47.g_dx
53$C_2^2$ \( 1 + 88 T^{2} + p^{2} T^{4} \) 2.53.a_dk
59$D_{4}$ \( 1 - 8 T + 126 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.59.ai_ew
61$D_{4}$ \( 1 + 18 T + 195 T^{2} + 18 p T^{3} + p^{2} T^{4} \) 2.61.s_hn
67$D_{4}$ \( 1 - 8 T + 142 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.67.ai_fm
71$D_{4}$ \( 1 - 20 T + 224 T^{2} - 20 p T^{3} + p^{2} T^{4} \) 2.71.au_iq
73$D_{4}$ \( 1 + 2 T - 53 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.73.c_acb
79$C_2^2$ \( 1 + 156 T^{2} + p^{2} T^{4} \) 2.79.a_ga
83$D_{4}$ \( 1 - 10 T + 173 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.83.ak_gr
89$D_{4}$ \( 1 + 20 T + 276 T^{2} + 20 p T^{3} + p^{2} T^{4} \) 2.89.u_kq
97$D_{4}$ \( 1 - 12 T + 222 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.97.am_io
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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

−7.955847194745948757466401449731, −7.49500824789736660604985527017, −7.07849168213727086924731594792, −6.73040059535711733973809028427, −6.63984849171206317592199153186, −6.22993282204430689929273626205, −5.74250233009827087768408143642, −5.41256067580243381231398531207, −4.98711705259485142364187464826, −4.80593901626654231046243005465, −4.57856578128829407279366795195, −4.45419551497814991871024807533, −3.59471129435026268497584483393, −3.40532119917113448227826210075, −2.81435699176931005484273141129, −2.62548876893381550671462056413, −2.19846638818811275826015644718, −1.72563003880105871263718893999, −1.05568521557433465559119258008, −0.45059304586768951778828774255, 0.45059304586768951778828774255, 1.05568521557433465559119258008, 1.72563003880105871263718893999, 2.19846638818811275826015644718, 2.62548876893381550671462056413, 2.81435699176931005484273141129, 3.40532119917113448227826210075, 3.59471129435026268497584483393, 4.45419551497814991871024807533, 4.57856578128829407279366795195, 4.80593901626654231046243005465, 4.98711705259485142364187464826, 5.41256067580243381231398531207, 5.74250233009827087768408143642, 6.22993282204430689929273626205, 6.63984849171206317592199153186, 6.73040059535711733973809028427, 7.07849168213727086924731594792, 7.49500824789736660604985527017, 7.955847194745948757466401449731

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