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Properties

Label 4-3528e2-1.1-c1e2-0-33
Degree $4$
Conductor $12446784$
Sign $1$
Analytic cond. $793.617$
Root an. cond. $5.30765$
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  + 2·5-s + 2·11-s + 4·13-s + 6·17-s + 4·19-s + 6·23-s + 5·25-s + 4·31-s − 10·37-s − 4·41-s − 8·43-s + 4·47-s − 12·53-s + 4·55-s + 12·59-s − 6·61-s + 8·65-s + 4·67-s + 28·71-s + 2·73-s + 8·79-s + 32·83-s + 12·85-s − 6·89-s + 8·95-s − 36·97-s + 14·101-s + ⋯
L(s)  = 1  + 0.894·5-s + 0.603·11-s + 1.10·13-s + 1.45·17-s + 0.917·19-s + 1.25·23-s + 25-s + 0.718·31-s − 1.64·37-s − 0.624·41-s − 1.21·43-s + 0.583·47-s − 1.64·53-s + 0.539·55-s + 1.56·59-s − 0.768·61-s + 0.992·65-s + 0.488·67-s + 3.32·71-s + 0.234·73-s + 0.900·79-s + 3.51·83-s + 1.30·85-s − 0.635·89-s + 0.820·95-s − 3.65·97-s + 1.39·101-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 12446784 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12446784 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(12446784\)    =    \(2^{6} \cdot 3^{4} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(793.617\)
Root analytic conductor: \(5.30765\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 12446784,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(5.252818746\)
\(L(\frac12)\) \(\approx\) \(5.252818746\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 \)
good5$C_2^2$ \( 1 - 2 T - T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.5.ac_ab
11$C_2^2$ \( 1 - 2 T - 7 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.11.ac_ah
13$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.13.ae_be
17$C_2^2$ \( 1 - 6 T + 19 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.17.ag_t
19$C_2^2$ \( 1 - 4 T - 3 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.19.ae_ad
23$C_2^2$ \( 1 - 6 T + 13 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.23.ag_n
29$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.29.a_cg
31$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.31.ae_ap
37$C_2$ \( ( 1 - T + p T^{2} )( 1 + 11 T + p T^{2} ) \) 2.37.k_cl
41$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.41.e_di
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.43.i_dy
47$C_2^2$ \( 1 - 4 T - 31 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.47.ae_abf
53$C_2^2$ \( 1 + 12 T + 91 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.53.m_dn
59$C_2^2$ \( 1 - 12 T + 85 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.59.am_dh
61$C_2^2$ \( 1 + 6 T - 25 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.61.g_az
67$C_2^2$ \( 1 - 4 T - 51 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.67.ae_abz
71$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \) 2.71.abc_na
73$C_2^2$ \( 1 - 2 T - 69 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.73.ac_acr
79$C_2^2$ \( 1 - 8 T - 15 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.79.ai_ap
83$C_2$ \( ( 1 - 16 T + p T^{2} )^{2} \) 2.83.abg_qg
89$C_2^2$ \( 1 + 6 T - 53 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.89.g_acb
97$C_2$ \( ( 1 + 18 T + p T^{2} )^{2} \) 2.97.bk_ty
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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

−8.594316877433638015503597470093, −8.579932130807695909402186706614, −8.031748499047061018190816522380, −7.74501740240241276984301594968, −7.12062199690342221179830772875, −6.87738901050974718518305273861, −6.44733966982346782127195980896, −6.31054832185485291617643433781, −5.67453872528464016156254880562, −5.35058274430357476400357544452, −4.92511632995996058074602031025, −4.91880135636564597255135759020, −3.93104965679820702167924511629, −3.61979417936196403827101944881, −3.18140890760127589043510436124, −3.02996417762507373136465266431, −2.08149274102353223279999162553, −1.78401865681617892675320519427, −0.994746337416827023848972097158, −0.895801900447360113478342261274, 0.895801900447360113478342261274, 0.994746337416827023848972097158, 1.78401865681617892675320519427, 2.08149274102353223279999162553, 3.02996417762507373136465266431, 3.18140890760127589043510436124, 3.61979417936196403827101944881, 3.93104965679820702167924511629, 4.91880135636564597255135759020, 4.92511632995996058074602031025, 5.35058274430357476400357544452, 5.67453872528464016156254880562, 6.31054832185485291617643433781, 6.44733966982346782127195980896, 6.87738901050974718518305273861, 7.12062199690342221179830772875, 7.74501740240241276984301594968, 8.031748499047061018190816522380, 8.579932130807695909402186706614, 8.594316877433638015503597470093

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