Properties

Label 4-2100e2-1.1-c3e2-0-13
Degree $4$
Conductor $4410000$
Sign $1$
Analytic cond. $15352.2$
Root an. cond. $11.1312$
Motivic weight $3$
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  − 6·3-s + 14·7-s + 27·9-s + 54·11-s + 74·13-s + 98·17-s + 42·19-s − 84·21-s + 58·23-s − 108·27-s + 248·29-s + 216·31-s − 324·33-s + 242·37-s − 444·39-s − 192·41-s + 24·43-s + 582·47-s + 147·49-s − 588·51-s − 112·53-s − 252·57-s − 746·59-s + 52·61-s + 378·63-s + 684·67-s − 348·69-s + ⋯
L(s)  = 1  − 1.15·3-s + 0.755·7-s + 9-s + 1.48·11-s + 1.57·13-s + 1.39·17-s + 0.507·19-s − 0.872·21-s + 0.525·23-s − 0.769·27-s + 1.58·29-s + 1.25·31-s − 1.70·33-s + 1.07·37-s − 1.82·39-s − 0.731·41-s + 0.0851·43-s + 1.80·47-s + 3/7·49-s − 1.61·51-s − 0.290·53-s − 0.585·57-s − 1.64·59-s + 0.109·61-s + 0.755·63-s + 1.24·67-s − 0.607·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4410000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410000 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(4410000\)    =    \(2^{4} \cdot 3^{2} \cdot 5^{4} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(15352.2\)
Root analytic conductor: \(11.1312\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 4410000,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(5.112166262\)
\(L(\frac12)\) \(\approx\) \(5.112166262\)
\(L(\frac{5}{2})\) 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$C_1$ \( ( 1 + p T )^{2} \)
5 \( 1 \)
7$C_1$ \( ( 1 - p T )^{2} \)
good11$D_{4}$ \( 1 - 54 T + 2955 T^{2} - 54 p^{3} T^{3} + p^{6} T^{4} \)
13$D_{4}$ \( 1 - 74 T + 3038 T^{2} - 74 p^{3} T^{3} + p^{6} T^{4} \)
17$D_{4}$ \( 1 - 98 T + 9502 T^{2} - 98 p^{3} T^{3} + p^{6} T^{4} \)
19$D_{4}$ \( 1 - 42 T + 5330 T^{2} - 42 p^{3} T^{3} + p^{6} T^{4} \)
23$D_{4}$ \( 1 - 58 T - 2729 T^{2} - 58 p^{3} T^{3} + p^{6} T^{4} \)
29$D_{4}$ \( 1 - 248 T + 61429 T^{2} - 248 p^{3} T^{3} + p^{6} T^{4} \)
31$D_{4}$ \( 1 - 216 T + 60346 T^{2} - 216 p^{3} T^{3} + p^{6} T^{4} \)
37$D_{4}$ \( 1 - 242 T + 80631 T^{2} - 242 p^{3} T^{3} + p^{6} T^{4} \)
41$D_{4}$ \( 1 + 192 T + 140082 T^{2} + 192 p^{3} T^{3} + p^{6} T^{4} \)
43$D_{4}$ \( 1 - 24 T + 9937 T^{2} - 24 p^{3} T^{3} + p^{6} T^{4} \)
47$D_{4}$ \( 1 - 582 T + 279138 T^{2} - 582 p^{3} T^{3} + p^{6} T^{4} \)
53$D_{4}$ \( 1 + 112 T + 265574 T^{2} + 112 p^{3} T^{3} + p^{6} T^{4} \)
59$D_{4}$ \( 1 + 746 T + 470426 T^{2} + 746 p^{3} T^{3} + p^{6} T^{4} \)
61$D_{4}$ \( 1 - 52 T + 433274 T^{2} - 52 p^{3} T^{3} + p^{6} T^{4} \)
67$D_{4}$ \( 1 - 684 T + 717509 T^{2} - 684 p^{3} T^{3} + p^{6} T^{4} \)
71$D_{4}$ \( 1 + 1010 T + 959947 T^{2} + 1010 p^{3} T^{3} + p^{6} T^{4} \)
73$D_{4}$ \( 1 + 758 T + 903254 T^{2} + 758 p^{3} T^{3} + p^{6} T^{4} \)
79$D_{4}$ \( 1 + 1420 T + 1465653 T^{2} + 1420 p^{3} T^{3} + p^{6} T^{4} \)
83$D_{4}$ \( 1 - 616 T + 104402 T^{2} - 616 p^{3} T^{3} + p^{6} T^{4} \)
89$D_{4}$ \( 1 - 390 T + 1434774 T^{2} - 390 p^{3} T^{3} + p^{6} T^{4} \)
97$D_{4}$ \( 1 - 824 T + 1520286 T^{2} - 824 p^{3} T^{3} + p^{6} T^{4} \)
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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

−8.839020637679438962438054246907, −8.660681346588510715851278661209, −8.028316523365771278888743096194, −7.922065897600999570356155061410, −7.24470093947686550662384877534, −6.97584779158917003164441571871, −6.43505295599236003113349769683, −6.21583090682654155982218549237, −5.70397031355003125619207722594, −5.63016141333174865729898076790, −4.75833103936539815706254760731, −4.69782393429834785439897408207, −4.02353898938027299224793155404, −3.86616723306911510718976498766, −3.03231072742430237417772370014, −2.82799271880609039780699697083, −1.69122191498879502439442344876, −1.32552411364965075653767605229, −1.00949073374241936151229301270, −0.63527393943777789908092387398, 0.63527393943777789908092387398, 1.00949073374241936151229301270, 1.32552411364965075653767605229, 1.69122191498879502439442344876, 2.82799271880609039780699697083, 3.03231072742430237417772370014, 3.86616723306911510718976498766, 4.02353898938027299224793155404, 4.69782393429834785439897408207, 4.75833103936539815706254760731, 5.63016141333174865729898076790, 5.70397031355003125619207722594, 6.21583090682654155982218549237, 6.43505295599236003113349769683, 6.97584779158917003164441571871, 7.24470093947686550662384877534, 7.922065897600999570356155061410, 8.028316523365771278888743096194, 8.660681346588510715851278661209, 8.839020637679438962438054246907

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