Properties

Label 4-112e2-1.1-c5e2-0-1
Degree $4$
Conductor $12544$
Sign $1$
Analytic cond. $322.668$
Root an. cond. $4.23827$
Motivic weight $5$
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  + 6·3-s − 18·5-s − 98·7-s + 54·9-s − 396·11-s − 350·13-s − 108·15-s + 1.80e3·17-s + 3.26e3·19-s − 588·21-s − 2.08e3·23-s − 4.58e3·25-s + 1.89e3·27-s + 6.69e3·29-s + 20·31-s − 2.37e3·33-s + 1.76e3·35-s + 6.23e3·37-s − 2.10e3·39-s − 6.04e3·41-s + 3.02e3·43-s − 972·45-s − 1.17e4·47-s + 7.20e3·49-s + 1.08e4·51-s + 9.46e3·53-s + 7.12e3·55-s + ⋯
L(s)  = 1  + 0.384·3-s − 0.321·5-s − 0.755·7-s + 2/9·9-s − 0.986·11-s − 0.574·13-s − 0.123·15-s + 1.51·17-s + 2.07·19-s − 0.290·21-s − 0.823·23-s − 1.46·25-s + 0.498·27-s + 1.47·29-s + 0.00373·31-s − 0.379·33-s + 0.243·35-s + 0.748·37-s − 0.221·39-s − 0.561·41-s + 0.249·43-s − 0.0715·45-s − 0.772·47-s + 3/7·49-s + 0.581·51-s + 0.462·53-s + 0.317·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 12544 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12544 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(12544\)    =    \(2^{8} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(322.668\)
Root analytic conductor: \(4.23827\)
Motivic weight: \(5\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{112} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 12544,\ (\ :5/2, 5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(1.786178390\)
\(L(\frac12)\) \(\approx\) \(1.786178390\)
\(L(\frac{7}{2})\) 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 \( 1 \)
7$C_1$ \( ( 1 + p^{2} T )^{2} \)
good3$D_{4}$ \( 1 - 2 p T - 2 p^{2} T^{2} - 2 p^{6} T^{3} + p^{10} T^{4} \)
5$D_{4}$ \( 1 + 18 T + 4906 T^{2} + 18 p^{5} T^{3} + p^{10} T^{4} \)
11$D_{4}$ \( 1 + 36 p T + 142198 T^{2} + 36 p^{6} T^{3} + p^{10} T^{4} \)
13$D_{4}$ \( 1 + 350 T + 546978 T^{2} + 350 p^{5} T^{3} + p^{10} T^{4} \)
17$D_{4}$ \( 1 - 1800 T + 3567406 T^{2} - 1800 p^{5} T^{3} + p^{10} T^{4} \)
19$D_{4}$ \( 1 - 3266 T + 7614270 T^{2} - 3266 p^{5} T^{3} + p^{10} T^{4} \)
23$D_{4}$ \( 1 + 2088 T + 9365230 T^{2} + 2088 p^{5} T^{3} + p^{10} T^{4} \)
29$D_{4}$ \( 1 - 6696 T + 51326470 T^{2} - 6696 p^{5} T^{3} + p^{10} T^{4} \)
31$D_{4}$ \( 1 - 20 T + 53103102 T^{2} - 20 p^{5} T^{3} + p^{10} T^{4} \)
37$D_{4}$ \( 1 - 6232 T + 144242070 T^{2} - 6232 p^{5} T^{3} + p^{10} T^{4} \)
41$D_{4}$ \( 1 + 6048 T + 223864366 T^{2} + 6048 p^{5} T^{3} + p^{10} T^{4} \)
43$D_{4}$ \( 1 - 3020 T - 30383466 T^{2} - 3020 p^{5} T^{3} + p^{10} T^{4} \)
47$D_{4}$ \( 1 + 11700 T + 292735582 T^{2} + 11700 p^{5} T^{3} + p^{10} T^{4} \)
53$D_{4}$ \( 1 - 9468 T + 858185230 T^{2} - 9468 p^{5} T^{3} + p^{10} T^{4} \)
59$D_{4}$ \( 1 - 43938 T + 1852599934 T^{2} - 43938 p^{5} T^{3} + p^{10} T^{4} \)
61$D_{4}$ \( 1 + 64754 T + 2408321418 T^{2} + 64754 p^{5} T^{3} + p^{10} T^{4} \)
67$D_{4}$ \( 1 + 24784 T + 2799959190 T^{2} + 24784 p^{5} T^{3} + p^{10} T^{4} \)
71$D_{4}$ \( 1 + 97416 T + 5729557966 T^{2} + 97416 p^{5} T^{3} + p^{10} T^{4} \)
73$D_{4}$ \( 1 - 17452 T + 3828622374 T^{2} - 17452 p^{5} T^{3} + p^{10} T^{4} \)
79$D_{4}$ \( 1 + 51256 T + 3645565854 T^{2} + 51256 p^{5} T^{3} + p^{10} T^{4} \)
83$D_{4}$ \( 1 + 117558 T + 7798161502 T^{2} + 117558 p^{5} T^{3} + p^{10} T^{4} \)
89$D_{4}$ \( 1 - 84276 T + 5915697430 T^{2} - 84276 p^{5} T^{3} + p^{10} T^{4} \)
97$D_{4}$ \( 1 - 20776 T + 16174049358 T^{2} - 20776 p^{5} T^{3} + p^{10} T^{4} \)
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

−13.02848526445948197856614961802, −12.30208050176208976936281176097, −11.81357685016776800200109602500, −11.80216158894445647133501736819, −10.65909771382784445835039782122, −10.07034810255686726169784431672, −9.852044866254439594247604416569, −9.448292660857559788389198491468, −8.440639671536460691143190297279, −8.027405614223930089162366475093, −7.41186882697124713525694691142, −7.15426099659702670297806941846, −5.98851350323583020859593161823, −5.64810768471317763730689485251, −4.83452450003376614175435518495, −4.02571392461900285695084184627, −3.00532177992967517781762274617, −2.93176159022537516546724942375, −1.53458039788581897718411694962, −0.50091096055834706358739441815, 0.50091096055834706358739441815, 1.53458039788581897718411694962, 2.93176159022537516546724942375, 3.00532177992967517781762274617, 4.02571392461900285695084184627, 4.83452450003376614175435518495, 5.64810768471317763730689485251, 5.98851350323583020859593161823, 7.15426099659702670297806941846, 7.41186882697124713525694691142, 8.027405614223930089162366475093, 8.440639671536460691143190297279, 9.448292660857559788389198491468, 9.852044866254439594247604416569, 10.07034810255686726169784431672, 10.65909771382784445835039782122, 11.80216158894445647133501736819, 11.81357685016776800200109602500, 12.30208050176208976936281176097, 13.02848526445948197856614961802

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