Properties

Label 4-9072e2-1.1-c1e2-0-0
Degree $4$
Conductor $82301184$
Sign $1$
Analytic cond. $5247.59$
Root an. cond. $8.51118$
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  − 4·5-s + 2·7-s + 2·11-s − 6·13-s − 14·17-s + 2·19-s + 2·23-s + 5·25-s − 10·29-s − 6·31-s − 8·35-s + 4·37-s − 12·41-s − 4·43-s + 6·47-s + 3·49-s + 12·53-s − 8·55-s − 2·59-s − 6·61-s + 24·65-s + 10·67-s + 20·71-s + 20·73-s + 4·77-s − 6·79-s + 2·83-s + ⋯
L(s)  = 1  − 1.78·5-s + 0.755·7-s + 0.603·11-s − 1.66·13-s − 3.39·17-s + 0.458·19-s + 0.417·23-s + 25-s − 1.85·29-s − 1.07·31-s − 1.35·35-s + 0.657·37-s − 1.87·41-s − 0.609·43-s + 0.875·47-s + 3/7·49-s + 1.64·53-s − 1.07·55-s − 0.260·59-s − 0.768·61-s + 2.97·65-s + 1.22·67-s + 2.37·71-s + 2.34·73-s + 0.455·77-s − 0.675·79-s + 0.219·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 82301184 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 82301184 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(82301184\)    =    \(2^{8} \cdot 3^{8} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(5247.59\)
Root analytic conductor: \(8.51118\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 82301184,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9213517168\)
\(L(\frac12)\) \(\approx\) \(0.9213517168\)
\(L(\frac{3}{2})\) 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)$
bad2 \( 1 \)
3 \( 1 \)
7$C_1$ \( ( 1 - T )^{2} \)
good5$C_2^2$ \( 1 + 4 T + 11 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 - 2 T - 4 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
13$C_2^2$ \( 1 + 6 T + 23 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
17$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
19$D_{4}$ \( 1 - 2 T + 36 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 - 2 T + 20 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 10 T + 71 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 + 6 T + 44 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 - 4 T + 3 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 + 12 T + 106 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 + 4 T + 78 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 - 6 T + 100 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 - 12 T + 130 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 2 T + 92 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 6 T + 83 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 - 10 T + 156 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 - 20 T + 230 T^{2} - 20 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 20 T + 243 T^{2} - 20 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 6 T + 20 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 2 T - 76 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 6 T + 139 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 8 T + 162 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
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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

−7.70558541584594560047568103350, −7.64203555595609987731472331670, −7.15846079584156871249242218828, −7.03181860490831556903116049342, −6.66187646580121934305481192263, −6.48357935773604818680963619290, −5.71151105207996987462752611948, −5.38606041963601383775408052984, −4.94383022498938847150377280937, −4.84234298560797510584726947305, −4.24501694857972456101879043245, −4.14017079774747966870773150680, −3.75458468968097826075802367168, −3.46412831478048506215546210114, −2.81341164175052098726897818785, −2.26762568290556402627350848754, −2.02253417661674526010801469183, −1.72550287893429206533658204257, −0.59895548263554099142157979052, −0.35038682991331631674479843579, 0.35038682991331631674479843579, 0.59895548263554099142157979052, 1.72550287893429206533658204257, 2.02253417661674526010801469183, 2.26762568290556402627350848754, 2.81341164175052098726897818785, 3.46412831478048506215546210114, 3.75458468968097826075802367168, 4.14017079774747966870773150680, 4.24501694857972456101879043245, 4.84234298560797510584726947305, 4.94383022498938847150377280937, 5.38606041963601383775408052984, 5.71151105207996987462752611948, 6.48357935773604818680963619290, 6.66187646580121934305481192263, 7.03181860490831556903116049342, 7.15846079584156871249242218828, 7.64203555595609987731472331670, 7.70558541584594560047568103350

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