Properties

Label 4-1620e2-1.1-c1e2-0-13
Degree $4$
Conductor $2624400$
Sign $1$
Analytic cond. $167.334$
Root an. cond. $3.59663$
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 + 4·13-s + 6·17-s − 2·19-s + 3·25-s + 12·29-s − 2·31-s − 4·35-s − 2·37-s + 12·41-s + 10·43-s + 6·47-s − 8·49-s + 18·53-s + 12·59-s − 8·61-s + 8·65-s − 8·67-s + 12·71-s + 10·73-s − 8·79-s + 6·83-s + 12·85-s − 8·91-s − 4·95-s − 2·97-s + ⋯
L(s)  = 1  + 0.894·5-s − 0.755·7-s + 1.10·13-s + 1.45·17-s − 0.458·19-s + 3/5·25-s + 2.22·29-s − 0.359·31-s − 0.676·35-s − 0.328·37-s + 1.87·41-s + 1.52·43-s + 0.875·47-s − 8/7·49-s + 2.47·53-s + 1.56·59-s − 1.02·61-s + 0.992·65-s − 0.977·67-s + 1.42·71-s + 1.17·73-s − 0.900·79-s + 0.658·83-s + 1.30·85-s − 0.838·91-s − 0.410·95-s − 0.203·97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2624400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2624400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2624400\)    =    \(2^{4} \cdot 3^{8} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(167.334\)
Root analytic conductor: \(3.59663\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1620} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 2624400,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.449660787\)
\(L(\frac12)\) \(\approx\) \(3.449660787\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5$C_1$ \( ( 1 - T )^{2} \)
good7$D_{4}$ \( 1 + 2 T + 12 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + 19 T^{2} + p^{2} T^{4} \)
13$D_{4}$ \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 - 6 T + 40 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 + 2 T + 27 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 + 34 T^{2} + p^{2} T^{4} \)
29$D_{4}$ \( 1 - 12 T + 91 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 + 2 T + 15 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 + 2 T + 48 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 - 12 T + 91 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - 10 T + 108 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 - 6 T + 100 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 - 18 T + 184 T^{2} - 18 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 - 12 T + 151 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
61$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
67$D_{4}$ \( 1 + 8 T + 42 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 - 12 T + 151 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 10 T + 144 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 8 T + 66 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 6 T + 28 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
89$C_2^2$ \( 1 + 151 T^{2} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 2 T + 192 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
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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

−9.445793774507647510215971479441, −9.326599821164464428207386751854, −8.779425679424916360269217981607, −8.571275760361312750626586983059, −7.971780380203900873912842064569, −7.69249645127935090704980304523, −7.07640845622787122576374735369, −6.75770147679716341484852819185, −6.17415727635653731346811227811, −6.09294264949099454088106951846, −5.52591078784078507001710964719, −5.32404448871283120218357725714, −4.44869409446023593015511679432, −4.25332733345007527784652413659, −3.36814822788499321950969048033, −3.36063942565164379448368190773, −2.41348115153317722710224549998, −2.28133981595779567490894279524, −1.05917541111005367081889955987, −0.932769269361741768893001655107, 0.932769269361741768893001655107, 1.05917541111005367081889955987, 2.28133981595779567490894279524, 2.41348115153317722710224549998, 3.36063942565164379448368190773, 3.36814822788499321950969048033, 4.25332733345007527784652413659, 4.44869409446023593015511679432, 5.32404448871283120218357725714, 5.52591078784078507001710964719, 6.09294264949099454088106951846, 6.17415727635653731346811227811, 6.75770147679716341484852819185, 7.07640845622787122576374735369, 7.69249645127935090704980304523, 7.971780380203900873912842064569, 8.571275760361312750626586983059, 8.779425679424916360269217981607, 9.326599821164464428207386751854, 9.445793774507647510215971479441

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