Properties

Label 4-1368e2-1.1-c3e2-0-0
Degree $4$
Conductor $1871424$
Sign $1$
Analytic cond. $6514.84$
Root an. cond. $8.98413$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 13·5-s + 3·7-s + 41·11-s + 8·13-s + 103·17-s + 38·19-s + 74·23-s − 99·25-s + 186·29-s − 30·31-s + 39·35-s − 274·37-s + 392·41-s − 193·43-s + 275·47-s − 461·49-s + 250·53-s + 533·55-s + 684·59-s + 201·61-s + 104·65-s − 384·67-s + 868·71-s + 715·73-s + 123·77-s + 420·79-s + 1.19e3·83-s + ⋯
L(s)  = 1  + 1.16·5-s + 0.161·7-s + 1.12·11-s + 0.170·13-s + 1.46·17-s + 0.458·19-s + 0.670·23-s − 0.791·25-s + 1.19·29-s − 0.173·31-s + 0.188·35-s − 1.21·37-s + 1.49·41-s − 0.684·43-s + 0.853·47-s − 1.34·49-s + 0.647·53-s + 1.30·55-s + 1.50·59-s + 0.421·61-s + 0.198·65-s − 0.700·67-s + 1.45·71-s + 1.14·73-s + 0.182·77-s + 0.598·79-s + 1.58·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1871424 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1871424 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1871424\)    =    \(2^{6} \cdot 3^{4} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(6514.84\)
Root analytic conductor: \(8.98413\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1871424,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(7.122725149\)
\(L(\frac12)\) \(\approx\) \(7.122725149\)
\(L(\frac{5}{2})\) 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 \( 1 \)
19$C_1$ \( ( 1 - p T )^{2} \)
good5$D_{4}$ \( 1 - 13 T + 268 T^{2} - 13 p^{3} T^{3} + p^{6} T^{4} \)
7$D_{4}$ \( 1 - 3 T + 470 T^{2} - 3 p^{3} T^{3} + p^{6} T^{4} \)
11$D_{4}$ \( 1 - 41 T + 2864 T^{2} - 41 p^{3} T^{3} + p^{6} T^{4} \)
13$D_{4}$ \( 1 - 8 T + 918 T^{2} - 8 p^{3} T^{3} + p^{6} T^{4} \)
17$D_{4}$ \( 1 - 103 T + 12260 T^{2} - 103 p^{3} T^{3} + p^{6} T^{4} \)
23$D_{4}$ \( 1 - 74 T + 17846 T^{2} - 74 p^{3} T^{3} + p^{6} T^{4} \)
29$D_{4}$ \( 1 - 186 T + 49570 T^{2} - 186 p^{3} T^{3} + p^{6} T^{4} \)
31$D_{4}$ \( 1 + 30 T + 31774 T^{2} + 30 p^{3} T^{3} + p^{6} T^{4} \)
37$D_{4}$ \( 1 + 274 T + 92042 T^{2} + 274 p^{3} T^{3} + p^{6} T^{4} \)
41$D_{4}$ \( 1 - 392 T + 137458 T^{2} - 392 p^{3} T^{3} + p^{6} T^{4} \)
43$D_{4}$ \( 1 + 193 T + 141918 T^{2} + 193 p^{3} T^{3} + p^{6} T^{4} \)
47$D_{4}$ \( 1 - 275 T + 225364 T^{2} - 275 p^{3} T^{3} + p^{6} T^{4} \)
53$D_{4}$ \( 1 - 250 T + 285346 T^{2} - 250 p^{3} T^{3} + p^{6} T^{4} \)
59$D_{4}$ \( 1 - 684 T + 480774 T^{2} - 684 p^{3} T^{3} + p^{6} T^{4} \)
61$D_{4}$ \( 1 - 201 T + 463456 T^{2} - 201 p^{3} T^{3} + p^{6} T^{4} \)
67$D_{4}$ \( 1 + 384 T + 599590 T^{2} + 384 p^{3} T^{3} + p^{6} T^{4} \)
71$D_{4}$ \( 1 - 868 T + 481646 T^{2} - 868 p^{3} T^{3} + p^{6} T^{4} \)
73$D_{4}$ \( 1 - 715 T + 596192 T^{2} - 715 p^{3} T^{3} + p^{6} T^{4} \)
79$D_{4}$ \( 1 - 420 T + 890110 T^{2} - 420 p^{3} T^{3} + p^{6} T^{4} \)
83$D_{4}$ \( 1 - 1198 T + 1305950 T^{2} - 1198 p^{3} T^{3} + p^{6} T^{4} \)
89$D_{4}$ \( 1 + 210 T + 784546 T^{2} + 210 p^{3} T^{3} + p^{6} T^{4} \)
97$D_{4}$ \( 1 - 328 T + 1569390 T^{2} - 328 p^{3} T^{3} + p^{6} T^{4} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.400885260455533615467408492963, −9.199582793945092539476116345059, −8.604630930429907773200487865443, −8.341003189952506804307015094749, −7.68758911997995461731966079021, −7.52955443202813479265148177343, −6.74420513352783352381936893881, −6.65745506141975721659074106884, −6.03528756632772767493033387193, −5.78273029603111952503714725232, −5.12454937982816581470582801825, −5.12113744818369471736053293847, −4.25484585950644526815049565636, −3.77272414368589463067622223064, −3.36239150650473712363279918444, −2.83566726831885896889895311848, −1.98312041724937916738305126333, −1.84814549456503108706977419379, −0.911666800159662167625397665418, −0.78456254464077430544065237861, 0.78456254464077430544065237861, 0.911666800159662167625397665418, 1.84814549456503108706977419379, 1.98312041724937916738305126333, 2.83566726831885896889895311848, 3.36239150650473712363279918444, 3.77272414368589463067622223064, 4.25484585950644526815049565636, 5.12113744818369471736053293847, 5.12454937982816581470582801825, 5.78273029603111952503714725232, 6.03528756632772767493033387193, 6.65745506141975721659074106884, 6.74420513352783352381936893881, 7.52955443202813479265148177343, 7.68758911997995461731966079021, 8.341003189952506804307015094749, 8.604630930429907773200487865443, 9.199582793945092539476116345059, 9.400885260455533615467408492963

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