Properties

Label 4-1960e2-1.1-c3e2-0-0
Degree $4$
Conductor $3841600$
Sign $1$
Analytic cond. $13373.4$
Root an. cond. $10.7537$
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  + 2·3-s − 10·5-s − 51·9-s + 26·11-s − 30·13-s − 20·15-s + 78·17-s − 112·19-s − 36·23-s + 75·25-s − 158·27-s − 66·29-s + 112·31-s + 52·33-s − 140·37-s − 60·39-s − 280·41-s − 40·43-s + 510·45-s − 366·47-s + 156·51-s + 164·53-s − 260·55-s − 224·57-s − 76·59-s + 932·61-s + 300·65-s + ⋯
L(s)  = 1  + 0.384·3-s − 0.894·5-s − 1.88·9-s + 0.712·11-s − 0.640·13-s − 0.344·15-s + 1.11·17-s − 1.35·19-s − 0.326·23-s + 3/5·25-s − 1.12·27-s − 0.422·29-s + 0.648·31-s + 0.274·33-s − 0.622·37-s − 0.246·39-s − 1.06·41-s − 0.141·43-s + 1.68·45-s − 1.13·47-s + 0.428·51-s + 0.425·53-s − 0.637·55-s − 0.520·57-s − 0.167·59-s + 1.95·61-s + 0.572·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3841600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3841600 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(3841600\)    =    \(2^{6} \cdot 5^{2} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(13373.4\)
Root analytic conductor: \(10.7537\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 3841600,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(1.429737499\)
\(L(\frac12)\) \(\approx\) \(1.429737499\)
\(L(\frac{5}{2})\) 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 \( 1 \)
5$C_1$ \( ( 1 + p T )^{2} \)
7 \( 1 \)
good3$C_2$ \( ( 1 - T + p^{3} T^{2} )^{2} \)
11$D_{4}$ \( 1 - 26 T + 155 T^{2} - 26 p^{3} T^{3} + p^{6} T^{4} \)
13$D_{4}$ \( 1 + 30 T + 1943 T^{2} + 30 p^{3} T^{3} + p^{6} T^{4} \)
17$D_{4}$ \( 1 - 78 T + 8671 T^{2} - 78 p^{3} T^{3} + p^{6} T^{4} \)
19$D_{4}$ \( 1 + 112 T + 14178 T^{2} + 112 p^{3} T^{3} + p^{6} T^{4} \)
23$D_{4}$ \( 1 + 36 T + 21982 T^{2} + 36 p^{3} T^{3} + p^{6} T^{4} \)
29$D_{4}$ \( 1 + 66 T + 39163 T^{2} + 66 p^{3} T^{3} + p^{6} T^{4} \)
31$D_{4}$ \( 1 - 112 T + 38634 T^{2} - 112 p^{3} T^{3} + p^{6} T^{4} \)
37$D_{4}$ \( 1 + 140 T + 103530 T^{2} + 140 p^{3} T^{3} + p^{6} T^{4} \)
41$D_{4}$ \( 1 + 280 T + 133358 T^{2} + 280 p^{3} T^{3} + p^{6} T^{4} \)
43$D_{4}$ \( 1 + 40 T + 135330 T^{2} + 40 p^{3} T^{3} + p^{6} T^{4} \)
47$D_{4}$ \( 1 + 366 T + 198319 T^{2} + 366 p^{3} T^{3} + p^{6} T^{4} \)
53$D_{4}$ \( 1 - 164 T + 208142 T^{2} - 164 p^{3} T^{3} + p^{6} T^{4} \)
59$D_{4}$ \( 1 + 76 T + 401498 T^{2} + 76 p^{3} T^{3} + p^{6} T^{4} \)
61$D_{4}$ \( 1 - 932 T + 604218 T^{2} - 932 p^{3} T^{3} + p^{6} T^{4} \)
67$D_{4}$ \( 1 + 196 T + 568314 T^{2} + 196 p^{3} T^{3} + p^{6} T^{4} \)
71$D_{4}$ \( 1 + 240 T + 558958 T^{2} + 240 p^{3} T^{3} + p^{6} T^{4} \)
73$D_{4}$ \( 1 - 132 T - 84634 T^{2} - 132 p^{3} T^{3} + p^{6} T^{4} \)
79$D_{4}$ \( 1 + 946 T + 757563 T^{2} + 946 p^{3} T^{3} + p^{6} T^{4} \)
83$D_{4}$ \( 1 - 704 T + 999878 T^{2} - 704 p^{3} T^{3} + p^{6} T^{4} \)
89$D_{4}$ \( 1 - 2020 T + 2298914 T^{2} - 2020 p^{3} T^{3} + p^{6} T^{4} \)
97$D_{4}$ \( 1 + 322 T + 1784367 T^{2} + 322 p^{3} T^{3} + p^{6} T^{4} \)
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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

−8.685585300887942771801319660144, −8.669234740047204164261435291301, −8.275799516972047856728562869556, −8.063392668487830720228623487953, −7.40296772815708214027961027650, −7.27224964570246633638556508895, −6.54435379165763244896589535821, −6.33490187348568648527733033420, −5.86032583581127545739050971791, −5.40112222344053816556616563835, −4.95064685684171685826495923869, −4.60657987874234581452182617920, −3.76966409135795542619349834018, −3.75232679957968136480950765478, −3.11936168932548934250870497506, −2.81903158490702813214074411909, −2.10234193864602974667423341796, −1.75488788657822735349079503938, −0.71116733305321872687782388520, −0.33690681663219090225550258540, 0.33690681663219090225550258540, 0.71116733305321872687782388520, 1.75488788657822735349079503938, 2.10234193864602974667423341796, 2.81903158490702813214074411909, 3.11936168932548934250870497506, 3.75232679957968136480950765478, 3.76966409135795542619349834018, 4.60657987874234581452182617920, 4.95064685684171685826495923869, 5.40112222344053816556616563835, 5.86032583581127545739050971791, 6.33490187348568648527733033420, 6.54435379165763244896589535821, 7.27224964570246633638556508895, 7.40296772815708214027961027650, 8.063392668487830720228623487953, 8.275799516972047856728562869556, 8.669234740047204164261435291301, 8.685585300887942771801319660144

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