Properties

Label 4-1216e2-1.1-c3e2-0-1
Degree $4$
Conductor $1478656$
Sign $1$
Analytic cond. $5147.53$
Root an. cond. $8.47032$
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  − 9·3-s + 9·5-s − 18·7-s + 25·9-s + 17·11-s − 17·13-s − 81·15-s − 80·17-s − 38·19-s + 162·21-s + 73·23-s − 25·25-s + 36·27-s − 3·29-s + 212·31-s − 153·33-s − 162·35-s − 192·37-s + 153·39-s − 50·41-s − 677·43-s + 225·45-s − 389·47-s − 151·49-s + 720·51-s + 1.21e3·53-s + 153·55-s + ⋯
L(s)  = 1  − 1.73·3-s + 0.804·5-s − 0.971·7-s + 0.925·9-s + 0.465·11-s − 0.362·13-s − 1.39·15-s − 1.14·17-s − 0.458·19-s + 1.68·21-s + 0.661·23-s − 1/5·25-s + 0.256·27-s − 0.0192·29-s + 1.22·31-s − 0.807·33-s − 0.782·35-s − 0.853·37-s + 0.628·39-s − 0.190·41-s − 2.40·43-s + 0.745·45-s − 1.20·47-s − 0.440·49-s + 1.97·51-s + 3.15·53-s + 0.375·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1478656 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1478656 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1478656\)    =    \(2^{12} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(5147.53\)
Root analytic conductor: \(8.47032\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1216} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1478656,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(1.135653441\)
\(L(\frac12)\) \(\approx\) \(1.135653441\)
\(L(\frac{5}{2})\) 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 \( 1 \)
19$C_1$ \( ( 1 + p T )^{2} \)
good3$D_{4}$ \( 1 + p^{2} T + 56 T^{2} + p^{5} T^{3} + p^{6} T^{4} \)
5$D_{4}$ \( 1 - 9 T + 106 T^{2} - 9 p^{3} T^{3} + p^{6} T^{4} \)
7$D_{4}$ \( 1 + 18 T + 475 T^{2} + 18 p^{3} T^{3} + p^{6} T^{4} \)
11$D_{4}$ \( 1 - 17 T + 2716 T^{2} - 17 p^{3} T^{3} + p^{6} T^{4} \)
13$D_{4}$ \( 1 + 17 T + 1382 T^{2} + 17 p^{3} T^{3} + p^{6} T^{4} \)
17$D_{4}$ \( 1 + 80 T + 11353 T^{2} + 80 p^{3} T^{3} + p^{6} T^{4} \)
23$D_{4}$ \( 1 - 73 T + 22582 T^{2} - 73 p^{3} T^{3} + p^{6} T^{4} \)
29$D_{4}$ \( 1 + 3 T + 40732 T^{2} + 3 p^{3} T^{3} + p^{6} T^{4} \)
31$D_{4}$ \( 1 - 212 T + 35486 T^{2} - 212 p^{3} T^{3} + p^{6} T^{4} \)
37$D_{4}$ \( 1 + 192 T + 96214 T^{2} + 192 p^{3} T^{3} + p^{6} T^{4} \)
41$D_{4}$ \( 1 + 50 T + 136642 T^{2} + 50 p^{3} T^{3} + p^{6} T^{4} \)
43$D_{4}$ \( 1 + 677 T + 272702 T^{2} + 677 p^{3} T^{3} + p^{6} T^{4} \)
47$D_{4}$ \( 1 + 389 T + 153478 T^{2} + 389 p^{3} T^{3} + p^{6} T^{4} \)
53$D_{4}$ \( 1 - 23 p T + 663970 T^{2} - 23 p^{4} T^{3} + p^{6} T^{4} \)
59$D_{4}$ \( 1 - 287 T + 419944 T^{2} - 287 p^{3} T^{3} + p^{6} T^{4} \)
61$D_{4}$ \( 1 + 313 T + 253596 T^{2} + 313 p^{3} T^{3} + p^{6} T^{4} \)
67$D_{4}$ \( 1 + 1223 T + 867254 T^{2} + 1223 p^{3} T^{3} + p^{6} T^{4} \)
71$D_{4}$ \( 1 - 200 T + 480250 T^{2} - 200 p^{3} T^{3} + p^{6} T^{4} \)
73$D_{4}$ \( 1 - 378 T + 195883 T^{2} - 378 p^{3} T^{3} + p^{6} T^{4} \)
79$D_{4}$ \( 1 - 1350 T + 1380310 T^{2} - 1350 p^{3} T^{3} + p^{6} T^{4} \)
83$D_{4}$ \( 1 - 670 T + 568942 T^{2} - 670 p^{3} T^{3} + p^{6} T^{4} \)
89$D_{4}$ \( 1 + 236 T + 778834 T^{2} + 236 p^{3} T^{3} + p^{6} T^{4} \)
97$D_{4}$ \( 1 - 1294 T + 2054082 T^{2} - 1294 p^{3} T^{3} + p^{6} T^{4} \)
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

−9.775358651900437767722628932714, −9.141176193706039641903660422111, −8.909466250171584697813728561853, −8.423507174570826249938716846313, −7.979741611359653850539559039884, −7.14621326168448255069470367164, −6.81893696166024585560969224083, −6.61862594344849791201250593154, −6.07300625482031912790381916961, −6.05806062086607062531733368008, −5.28687096676857247266586368394, −5.08304949754886314717530113192, −4.60335072780410334854389832680, −4.06706543697310112662461456778, −3.28512318185885222853574877961, −2.99673260301974631476974528511, −1.99477569822580676860643888500, −1.84338025992264241344640002805, −0.61572355752188987972237371952, −0.47450578254244322611080032702, 0.47450578254244322611080032702, 0.61572355752188987972237371952, 1.84338025992264241344640002805, 1.99477569822580676860643888500, 2.99673260301974631476974528511, 3.28512318185885222853574877961, 4.06706543697310112662461456778, 4.60335072780410334854389832680, 5.08304949754886314717530113192, 5.28687096676857247266586368394, 6.05806062086607062531733368008, 6.07300625482031912790381916961, 6.61862594344849791201250593154, 6.81893696166024585560969224083, 7.14621326168448255069470367164, 7.979741611359653850539559039884, 8.423507174570826249938716846313, 8.909466250171584697813728561853, 9.141176193706039641903660422111, 9.775358651900437767722628932714

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