Properties

Label 4-464e2-1.1-c3e2-0-2
Degree $4$
Conductor $215296$
Sign $1$
Analytic cond. $749.493$
Root an. cond. $5.23229$
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  + 10·3-s − 10·5-s + 16·7-s + 39·9-s + 26·11-s − 26·13-s − 100·15-s + 60·17-s + 220·19-s + 160·21-s − 52·23-s − 143·25-s + 50·27-s + 58·29-s + 294·31-s + 260·33-s − 160·35-s + 312·37-s − 260·39-s + 40·41-s + 322·43-s − 390·45-s + 130·47-s − 294·49-s + 600·51-s + 1.00e3·53-s − 260·55-s + ⋯
L(s)  = 1  + 1.92·3-s − 0.894·5-s + 0.863·7-s + 13/9·9-s + 0.712·11-s − 0.554·13-s − 1.72·15-s + 0.856·17-s + 2.65·19-s + 1.66·21-s − 0.471·23-s − 1.14·25-s + 0.356·27-s + 0.371·29-s + 1.70·31-s + 1.37·33-s − 0.772·35-s + 1.38·37-s − 1.06·39-s + 0.152·41-s + 1.14·43-s − 1.29·45-s + 0.403·47-s − 6/7·49-s + 1.64·51-s + 2.59·53-s − 0.637·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 215296 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 215296 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(215296\)    =    \(2^{8} \cdot 29^{2}\)
Sign: $1$
Analytic conductor: \(749.493\)
Root analytic conductor: \(5.23229\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{464} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 215296,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(6.780033548\)
\(L(\frac12)\) \(\approx\) \(6.780033548\)
\(L(\frac{5}{2})\) 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 \( 1 \)
29$C_1$ \( ( 1 - p T )^{2} \)
good3$D_{4}$ \( 1 - 10 T + 61 T^{2} - 10 p^{3} T^{3} + p^{6} T^{4} \)
5$D_{4}$ \( 1 + 2 p T + 243 T^{2} + 2 p^{4} T^{3} + p^{6} T^{4} \)
7$D_{4}$ \( 1 - 16 T + 550 T^{2} - 16 p^{3} T^{3} + p^{6} T^{4} \)
11$D_{4}$ \( 1 - 26 T + 93 T^{2} - 26 p^{3} T^{3} + p^{6} T^{4} \)
13$D_{4}$ \( 1 + 2 p T + 19 p^{2} T^{2} + 2 p^{4} T^{3} + p^{6} T^{4} \)
17$D_{4}$ \( 1 - 60 T + 10078 T^{2} - 60 p^{3} T^{3} + p^{6} T^{4} \)
19$D_{4}$ \( 1 - 220 T + 23770 T^{2} - 220 p^{3} T^{3} + p^{6} T^{4} \)
23$D_{4}$ \( 1 + 52 T + 20402 T^{2} + 52 p^{3} T^{3} + p^{6} T^{4} \)
31$D_{4}$ \( 1 - 294 T + 2363 p T^{2} - 294 p^{3} T^{3} + p^{6} T^{4} \)
37$D_{4}$ \( 1 - 312 T + 119370 T^{2} - 312 p^{3} T^{3} + p^{6} T^{4} \)
41$D_{4}$ \( 1 - 40 T + 100154 T^{2} - 40 p^{3} T^{3} + p^{6} T^{4} \)
43$D_{4}$ \( 1 - 322 T + 126453 T^{2} - 322 p^{3} T^{3} + p^{6} T^{4} \)
47$D_{4}$ \( 1 - 130 T + 126173 T^{2} - 130 p^{3} T^{3} + p^{6} T^{4} \)
53$D_{4}$ \( 1 - 1002 T + 518987 T^{2} - 1002 p^{3} T^{3} + p^{6} T^{4} \)
59$D_{4}$ \( 1 - 900 T + 490250 T^{2} - 900 p^{3} T^{3} + p^{6} T^{4} \)
61$D_{4}$ \( 1 + 948 T + 615270 T^{2} + 948 p^{3} T^{3} + p^{6} T^{4} \)
67$D_{4}$ \( 1 + 320 T + 158614 T^{2} + 320 p^{3} T^{3} + p^{6} T^{4} \)
71$D_{4}$ \( 1 - 660 T + 822410 T^{2} - 660 p^{3} T^{3} + p^{6} T^{4} \)
73$D_{4}$ \( 1 - 648 T + 63810 T^{2} - 648 p^{3} T^{3} + p^{6} T^{4} \)
79$D_{4}$ \( 1 + 258 T + 770157 T^{2} + 258 p^{3} T^{3} + p^{6} T^{4} \)
83$D_{4}$ \( 1 + 1212 T + 1502618 T^{2} + 1212 p^{3} T^{3} + p^{6} T^{4} \)
89$D_{4}$ \( 1 - 760 T + 1009370 T^{2} - 760 p^{3} T^{3} + p^{6} T^{4} \)
97$D_{4}$ \( 1 - 24 T + 1157322 T^{2} - 24 p^{3} T^{3} + p^{6} T^{4} \)
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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

−10.95583030356924640425444390547, −10.07555892973262500665317721605, −9.723173678106536532632278193007, −9.617928620828343113349283360001, −8.954385336525583748073464215368, −8.481707463687013210237351977198, −8.017236876011354319051527024498, −7.918083194474415170445556792178, −7.28051258209038325958056998714, −7.25999113088945968858502194409, −6.13289769561649529171874232601, −5.66891098212242015395114944740, −4.95550350079299034817694972694, −4.41824193807745198649569365538, −3.68264483548214855602328653688, −3.58581447039045162228379826295, −2.57973953734593212920978969947, −2.56075905492001820411926054467, −1.34340706674633016349526612836, −0.826075791967159972326976804706, 0.826075791967159972326976804706, 1.34340706674633016349526612836, 2.56075905492001820411926054467, 2.57973953734593212920978969947, 3.58581447039045162228379826295, 3.68264483548214855602328653688, 4.41824193807745198649569365538, 4.95550350079299034817694972694, 5.66891098212242015395114944740, 6.13289769561649529171874232601, 7.25999113088945968858502194409, 7.28051258209038325958056998714, 7.918083194474415170445556792178, 8.017236876011354319051527024498, 8.481707463687013210237351977198, 8.954385336525583748073464215368, 9.617928620828343113349283360001, 9.723173678106536532632278193007, 10.07555892973262500665317721605, 10.95583030356924640425444390547

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