Properties

Label 4-1216e2-1.1-c2e2-0-4
Degree $4$
Conductor $1478656$
Sign $1$
Analytic cond. $1097.83$
Root an. cond. $5.75617$
Motivic weight $2$
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  + 2·5-s − 10·7-s + 10·9-s + 10·11-s − 50·17-s + 38·19-s + 20·23-s − 47·25-s − 20·35-s + 10·43-s + 20·45-s − 10·47-s − 23·49-s + 20·55-s − 190·61-s − 100·63-s − 50·73-s − 100·77-s + 19·81-s − 260·83-s − 100·85-s + 76·95-s + 100·99-s + 40·115-s + 500·119-s − 167·121-s − 146·125-s + ⋯
L(s)  = 1  + 2/5·5-s − 1.42·7-s + 10/9·9-s + 0.909·11-s − 2.94·17-s + 2·19-s + 0.869·23-s − 1.87·25-s − 4/7·35-s + 0.232·43-s + 4/9·45-s − 0.212·47-s − 0.469·49-s + 4/11·55-s − 3.11·61-s − 1.58·63-s − 0.684·73-s − 1.29·77-s + 0.234·81-s − 3.13·83-s − 1.17·85-s + 4/5·95-s + 1.01·99-s + 8/23·115-s + 4.20·119-s − 1.38·121-s − 1.16·125-s + ⋯

Functional equation

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

Invariants

Degree: \(4\)
Conductor: \(1478656\)    =    \(2^{12} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(1097.83\)
Root analytic conductor: \(5.75617\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1216} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1478656,\ (\ :1, 1),\ 1)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.9955369302\)
\(L(\frac12)\) \(\approx\) \(0.9955369302\)
\(L(2)\) 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$C_2^2$ \( 1 - 10 T^{2} + p^{4} T^{4} \)
5$C_2$ \( ( 1 - T + p^{2} T^{2} )^{2} \)
7$C_2$ \( ( 1 + 5 T + p^{2} T^{2} )^{2} \)
11$C_2$ \( ( 1 - 5 T + p^{2} T^{2} )^{2} \)
13$C_2^2$ \( 1 - 50 T^{2} + p^{4} T^{4} \)
17$C_2$ \( ( 1 + 25 T + p^{2} T^{2} )^{2} \)
23$C_2$ \( ( 1 - 10 T + p^{2} T^{2} )^{2} \)
29$C_2^2$ \( 1 + 118 T^{2} + p^{4} T^{4} \)
31$C_2^2$ \( 1 - 122 T^{2} + p^{4} T^{4} \)
37$C_2^2$ \( 1 - 2090 T^{2} + p^{4} T^{4} \)
41$C_2^2$ \( 1 - 1562 T^{2} + p^{4} T^{4} \)
43$C_2$ \( ( 1 - 5 T + p^{2} T^{2} )^{2} \)
47$C_2$ \( ( 1 + 5 T + p^{2} T^{2} )^{2} \)
53$C_2^2$ \( 1 - 4970 T^{2} + p^{4} T^{4} \)
59$C_2$ \( ( 1 - 82 T + p^{2} T^{2} )( 1 + 82 T + p^{2} T^{2} ) \)
61$C_2$ \( ( 1 + 95 T + p^{2} T^{2} )^{2} \)
67$C_2^2$ \( 1 + 3190 T^{2} + p^{4} T^{4} \)
71$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
73$C_2$ \( ( 1 + 25 T + p^{2} T^{2} )^{2} \)
79$C_2^2$ \( 1 - 10682 T^{2} + p^{4} T^{4} \)
83$C_2$ \( ( 1 + 130 T + p^{2} T^{2} )^{2} \)
89$C_2^2$ \( 1 + 358 T^{2} + p^{4} T^{4} \)
97$C_2^2$ \( 1 - 18530 T^{2} + p^{4} T^{4} \)
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

−9.808961051413855264507718860053, −9.271442523655579667178009052352, −9.248721698815674170275888841848, −8.714721162518727063888217908963, −8.126668284886386582934411439794, −7.44798880492121281146743714858, −7.27818463600010053571397630899, −6.66607873069851729454915424837, −6.63279488846686857183789810806, −6.02618774718221267691993011872, −5.74087437482306725756663167257, −4.94968162690077241625855272886, −4.54049235092238358804179232279, −4.07753423488268528162614588511, −3.69752102300707023834843720612, −2.95769784078299026426650093159, −2.67701922849005728523121880777, −1.61890187183794563666153029922, −1.51464852629502267264975763343, −0.28877735576627156860567067410, 0.28877735576627156860567067410, 1.51464852629502267264975763343, 1.61890187183794563666153029922, 2.67701922849005728523121880777, 2.95769784078299026426650093159, 3.69752102300707023834843720612, 4.07753423488268528162614588511, 4.54049235092238358804179232279, 4.94968162690077241625855272886, 5.74087437482306725756663167257, 6.02618774718221267691993011872, 6.63279488846686857183789810806, 6.66607873069851729454915424837, 7.27818463600010053571397630899, 7.44798880492121281146743714858, 8.126668284886386582934411439794, 8.714721162518727063888217908963, 9.248721698815674170275888841848, 9.271442523655579667178009052352, 9.808961051413855264507718860053

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