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Properties

Label 4-8112e2-1.1-c1e2-0-9
Degree $4$
Conductor $65804544$
Sign $1$
Analytic cond. $4195.75$
Root an. cond. $8.04826$
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  + 2·3-s + 6·7-s + 3·9-s + 6·11-s + 6·19-s + 12·21-s + 6·23-s − 7·25-s + 4·27-s − 6·29-s + 12·31-s + 12·33-s − 6·37-s − 6·41-s − 2·43-s − 6·47-s + 16·49-s + 6·53-s + 12·57-s + 20·61-s + 18·63-s + 18·67-s + 12·69-s + 6·71-s − 14·75-s + 36·77-s + 4·79-s + ⋯
L(s)  = 1  + 1.15·3-s + 2.26·7-s + 9-s + 1.80·11-s + 1.37·19-s + 2.61·21-s + 1.25·23-s − 7/5·25-s + 0.769·27-s − 1.11·29-s + 2.15·31-s + 2.08·33-s − 0.986·37-s − 0.937·41-s − 0.304·43-s − 0.875·47-s + 16/7·49-s + 0.824·53-s + 1.58·57-s + 2.56·61-s + 2.26·63-s + 2.19·67-s + 1.44·69-s + 0.712·71-s − 1.61·75-s + 4.10·77-s + 0.450·79-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 65804544 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65804544 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(65804544\)    =    \(2^{8} \cdot 3^{2} \cdot 13^{4}\)
Sign: $1$
Analytic conductor: \(4195.75\)
Root analytic conductor: \(8.04826\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 65804544,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(11.91260270\)
\(L(\frac12)\) \(\approx\) \(11.91260270\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{2} \)
13 \( 1 \)
good5$C_2^2$ \( 1 + 7 T^{2} + p^{2} T^{4} \) 2.5.a_h
7$D_{4}$ \( 1 - 6 T + 20 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.7.ag_u
11$D_{4}$ \( 1 - 6 T + 28 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.11.ag_bc
17$C_2^2$ \( 1 + 7 T^{2} + p^{2} T^{4} \) 2.17.a_h
19$D_{4}$ \( 1 - 6 T + 44 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.19.ag_bs
23$D_{4}$ \( 1 - 6 T + 28 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.23.ag_bc
29$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.29.g_cp
31$D_{4}$ \( 1 - 12 T + 86 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.31.am_di
37$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.37.g_df
41$D_{4}$ \( 1 + 6 T + 79 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.41.g_db
43$D_{4}$ \( 1 + 2 T + 60 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.43.c_ci
47$D_{4}$ \( 1 + 6 T + 100 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.47.g_dw
53$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.53.ag_el
59$C_2^2$ \( 1 - 74 T^{2} + p^{2} T^{4} \) 2.59.a_acw
61$D_{4}$ \( 1 - 20 T + 195 T^{2} - 20 p T^{3} + p^{2} T^{4} \) 2.61.au_hn
67$D_{4}$ \( 1 - 18 T + 212 T^{2} - 18 p T^{3} + p^{2} T^{4} \) 2.67.as_ie
71$D_{4}$ \( 1 - 6 T + 124 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.71.ag_eu
73$C_2^2$ \( 1 - T^{2} + p^{2} T^{4} \) 2.73.a_ab
79$D_{4}$ \( 1 - 4 T + 54 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.79.ae_cc
83$D_{4}$ \( 1 + 6 T + 100 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.83.g_dw
89$D_{4}$ \( 1 + 12 T + 202 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.89.m_hu
97$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.97.m_iw
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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

−8.162700234538168287645368291621, −7.84215613085463274383807646135, −7.30439246734160862154266823515, −7.01674777910065709162708666010, −6.66405823675937687722773640641, −6.63935964434858668329060270853, −5.63356764568546511696116060816, −5.60946288208943413687365536257, −5.01971651818715916071095278206, −4.95973678291212891176733660879, −4.22635523271678249761150145422, −4.21224872492551556344528850037, −3.69662803564853256499726269575, −3.34503951987688547391178257593, −2.97073895090124355293024008149, −2.26442780187223413441318997649, −1.93140832918961304608985458014, −1.65233908017883234946768246486, −1.08786962914024383321329375324, −0.858194730371261638386845441323, 0.858194730371261638386845441323, 1.08786962914024383321329375324, 1.65233908017883234946768246486, 1.93140832918961304608985458014, 2.26442780187223413441318997649, 2.97073895090124355293024008149, 3.34503951987688547391178257593, 3.69662803564853256499726269575, 4.21224872492551556344528850037, 4.22635523271678249761150145422, 4.95973678291212891176733660879, 5.01971651818715916071095278206, 5.60946288208943413687365536257, 5.63356764568546511696116060816, 6.63935964434858668329060270853, 6.66405823675937687722773640641, 7.01674777910065709162708666010, 7.30439246734160862154266823515, 7.84215613085463274383807646135, 8.162700234538168287645368291621

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