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Properties

Label 4-2790e2-1.1-c1e2-0-3
Degree $4$
Conductor $7784100$
Sign $1$
Analytic cond. $496.320$
Root an. cond. $4.71998$
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  − 2·2-s + 3·4-s + 2·5-s − 7-s − 4·8-s − 4·10-s − 5·11-s + 12·13-s + 2·14-s + 5·16-s + 8·17-s + 19-s + 6·20-s + 10·22-s − 5·23-s + 3·25-s − 24·26-s − 3·28-s + 2·31-s − 6·32-s − 16·34-s − 2·35-s + 2·37-s − 2·38-s − 8·40-s + 2·41-s − 9·43-s + ⋯
L(s)  = 1  − 1.41·2-s + 3/2·4-s + 0.894·5-s − 0.377·7-s − 1.41·8-s − 1.26·10-s − 1.50·11-s + 3.32·13-s + 0.534·14-s + 5/4·16-s + 1.94·17-s + 0.229·19-s + 1.34·20-s + 2.13·22-s − 1.04·23-s + 3/5·25-s − 4.70·26-s − 0.566·28-s + 0.359·31-s − 1.06·32-s − 2.74·34-s − 0.338·35-s + 0.328·37-s − 0.324·38-s − 1.26·40-s + 0.312·41-s − 1.37·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7784100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7784100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(7784100\)    =    \(2^{2} \cdot 3^{4} \cdot 5^{2} \cdot 31^{2}\)
Sign: $1$
Analytic conductor: \(496.320\)
Root analytic conductor: \(4.71998\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 7784100,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.818669048\)
\(L(\frac12)\) \(\approx\) \(1.818669048\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{2} \)
3 \( 1 \)
5$C_1$ \( ( 1 - T )^{2} \)
31$C_1$ \( ( 1 - T )^{2} \)
good7$D_{4}$ \( 1 + T - 2 T^{2} + p T^{3} + p^{2} T^{4} \) 2.7.b_ac
11$D_{4}$ \( 1 + 5 T + 12 T^{2} + 5 p T^{3} + p^{2} T^{4} \) 2.11.f_m
13$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.13.am_ck
17$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.17.ai_by
19$D_{4}$ \( 1 - T + 22 T^{2} - p T^{3} + p^{2} T^{4} \) 2.19.ab_w
23$D_{4}$ \( 1 + 5 T + 36 T^{2} + 5 p T^{3} + p^{2} T^{4} \) 2.23.f_bk
29$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.29.a_cg
37$D_{4}$ \( 1 - 2 T + 10 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.37.ac_k
41$D_{4}$ \( 1 - 2 T + 18 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.41.ac_s
43$D_{4}$ \( 1 + 9 T + 90 T^{2} + 9 p T^{3} + p^{2} T^{4} \) 2.43.j_dm
47$D_{4}$ \( 1 + 6 T + 38 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.47.g_bm
53$D_{4}$ \( 1 + 3 T + 92 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.53.d_do
59$D_{4}$ \( 1 - 2 T + 54 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.59.ac_cc
61$D_{4}$ \( 1 + 6 T + 66 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.61.g_co
67$D_{4}$ \( 1 + 6 T + 78 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.67.g_da
71$D_{4}$ \( 1 - 17 T + 198 T^{2} - 17 p T^{3} + p^{2} T^{4} \) 2.71.ar_hq
73$D_{4}$ \( 1 + 9 T + 150 T^{2} + 9 p T^{3} + p^{2} T^{4} \) 2.73.j_fu
79$D_{4}$ \( 1 + 9 T + 162 T^{2} + 9 p T^{3} + p^{2} T^{4} \) 2.79.j_gg
83$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.83.aq_iw
89$D_{4}$ \( 1 - 5 T + 168 T^{2} - 5 p T^{3} + p^{2} T^{4} \) 2.89.af_gm
97$C_2^2$ \( 1 - 66 T^{2} + p^{2} T^{4} \) 2.97.a_aco
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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

−8.805598378178777518083228619532, −8.748764625514085868370584798305, −8.184430508147482061749939701546, −8.051302926031878531550217920613, −7.64058531780816519101915685917, −7.35145870006563386466186940461, −6.54233965340957264836063288495, −6.36081126922860334700090049837, −6.02826560522312880660647924233, −5.80545879091466943363675361482, −5.31241979353152848698695725299, −4.93981276658805860548767706815, −4.05651659749537333518747540161, −3.55965533248184233470433661693, −3.17110080301766094079044645077, −2.95168776743297228734946623259, −2.05951540710162360912851985623, −1.69779465124660635776341477032, −1.13112750137493544231039460021, −0.62849609232271989998880696885, 0.62849609232271989998880696885, 1.13112750137493544231039460021, 1.69779465124660635776341477032, 2.05951540710162360912851985623, 2.95168776743297228734946623259, 3.17110080301766094079044645077, 3.55965533248184233470433661693, 4.05651659749537333518747540161, 4.93981276658805860548767706815, 5.31241979353152848698695725299, 5.80545879091466943363675361482, 6.02826560522312880660647924233, 6.36081126922860334700090049837, 6.54233965340957264836063288495, 7.35145870006563386466186940461, 7.64058531780816519101915685917, 8.051302926031878531550217920613, 8.184430508147482061749939701546, 8.748764625514085868370584798305, 8.805598378178777518083228619532

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