Properties

Label 4-4032e2-1.1-c1e2-0-9
Degree $4$
Conductor $16257024$
Sign $1$
Analytic cond. $1036.56$
Root an. cond. $5.67412$
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 + 2·7-s − 2·11-s + 6·17-s + 4·19-s + 2·23-s − 2·25-s − 12·29-s + 4·31-s − 4·35-s + 10·41-s + 8·43-s − 12·47-s + 3·49-s − 16·53-s + 4·55-s − 4·59-s + 16·67-s − 10·71-s − 12·73-s − 4·77-s + 8·79-s + 8·83-s − 12·85-s + 18·89-s − 8·95-s + 4·97-s + ⋯
L(s)  = 1  − 0.894·5-s + 0.755·7-s − 0.603·11-s + 1.45·17-s + 0.917·19-s + 0.417·23-s − 2/5·25-s − 2.22·29-s + 0.718·31-s − 0.676·35-s + 1.56·41-s + 1.21·43-s − 1.75·47-s + 3/7·49-s − 2.19·53-s + 0.539·55-s − 0.520·59-s + 1.95·67-s − 1.18·71-s − 1.40·73-s − 0.455·77-s + 0.900·79-s + 0.878·83-s − 1.30·85-s + 1.90·89-s − 0.820·95-s + 0.406·97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16257024 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16257024 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(16257024\)    =    \(2^{12} \cdot 3^{4} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(1036.56\)
Root analytic conductor: \(5.67412\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 16257024,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.043324897\)
\(L(\frac12)\) \(\approx\) \(2.043324897\)
\(L(\frac{3}{2})\) 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$D_{4}$ \( 1 + 2 T + 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 + 2 T + 18 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
13$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \)
17$D_{4}$ \( 1 - 6 T + 38 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 - 4 T + 22 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
29$C_4$ \( 1 + 12 T + 74 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
31$C_4$ \( 1 - 4 T + 46 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
37$C_2^2$ \( 1 + 54 T^{2} + p^{2} T^{4} \)
41$D_{4}$ \( 1 - 10 T + 102 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - 8 T + 22 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + 12 T + 110 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
53$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
59$D_{4}$ \( 1 + 4 T + 102 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 102 T^{2} + p^{2} T^{4} \)
67$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
71$D_{4}$ \( 1 + 10 T + 162 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 12 T + 102 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - 8 T + 94 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 8 T + 102 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 - 18 T + 214 T^{2} - 18 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 4 T + 118 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
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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

−8.397894078659202842374292271942, −8.141251195093814845743144337126, −7.74803629948086968892868501982, −7.67935623269830673031702244961, −7.39706405776710828132755642144, −7.00666250163202302348594493315, −6.23139843744444886193257183897, −6.12028431514149768447169474846, −5.49792490397429553454540232845, −5.35377636814772194992335322770, −4.85948532410234309504798937827, −4.50759498393032602666360601197, −4.03151383480392728802042178548, −3.61143218202297850313775092033, −3.16208837271191926975435998809, −2.94454688541218234437712065114, −2.11744002994502970216774838265, −1.73404047467765986412029430117, −1.06806540269678381521730297541, −0.46714994894294464051440793949, 0.46714994894294464051440793949, 1.06806540269678381521730297541, 1.73404047467765986412029430117, 2.11744002994502970216774838265, 2.94454688541218234437712065114, 3.16208837271191926975435998809, 3.61143218202297850313775092033, 4.03151383480392728802042178548, 4.50759498393032602666360601197, 4.85948532410234309504798937827, 5.35377636814772194992335322770, 5.49792490397429553454540232845, 6.12028431514149768447169474846, 6.23139843744444886193257183897, 7.00666250163202302348594493315, 7.39706405776710828132755642144, 7.67935623269830673031702244961, 7.74803629948086968892868501982, 8.141251195093814845743144337126, 8.397894078659202842374292271942

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