Properties

Label 4-1134e2-1.1-c1e2-0-24
Degree $4$
Conductor $1285956$
Sign $1$
Analytic cond. $81.9936$
Root an. cond. $3.00915$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 4-s − 2·7-s + 16-s + 6·19-s + 2·25-s + 2·28-s + 6·31-s − 10·37-s − 10·43-s − 3·49-s + 12·61-s − 64-s − 2·67-s − 6·73-s − 6·76-s − 14·79-s + 30·97-s − 2·100-s + 24·103-s + 14·109-s − 2·112-s − 17·121-s − 6·124-s + 127-s + 131-s − 12·133-s + 137-s + ⋯
L(s)  = 1  − 1/2·4-s − 0.755·7-s + 1/4·16-s + 1.37·19-s + 2/5·25-s + 0.377·28-s + 1.07·31-s − 1.64·37-s − 1.52·43-s − 3/7·49-s + 1.53·61-s − 1/8·64-s − 0.244·67-s − 0.702·73-s − 0.688·76-s − 1.57·79-s + 3.04·97-s − 1/5·100-s + 2.36·103-s + 1.34·109-s − 0.188·112-s − 1.54·121-s − 0.538·124-s + 0.0887·127-s + 0.0873·131-s − 1.04·133-s + 0.0854·137-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1285956 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1285956 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1285956\)    =    \(2^{2} \cdot 3^{8} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(81.9936\)
Root analytic conductor: \(3.00915\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1285956,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.359733393\)
\(L(\frac12)\) \(\approx\) \(1.359733393\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T^{2} \)
3 \( 1 \)
7$C_2$ \( 1 + 2 T + p T^{2} \)
good5$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + 17 T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
17$C_2^2$ \( 1 - 11 T^{2} + p^{2} T^{4} \)
19$C_2$$\times$$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 - T + p T^{2} ) \)
23$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \)
31$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
37$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
41$C_2^2$ \( 1 + 13 T^{2} + p^{2} T^{4} \)
43$C_2$$\times$$C_2$ \( ( 1 - T + p T^{2} )( 1 + 11 T + p T^{2} ) \)
47$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \)
59$C_2^2$ \( 1 + T^{2} + p^{2} T^{4} \)
61$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 - 2 T + p T^{2} ) \)
67$C_2$$\times$$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 13 T + p T^{2} ) \)
71$C_2^2$ \( 1 + 110 T^{2} + p^{2} T^{4} \)
73$C_2$$\times$$C_2$ \( ( 1 + T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
79$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
83$C_2^2$ \( 1 + 106 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
97$C_2$$\times$$C_2$ \( ( 1 - 17 T + p T^{2} )( 1 - 13 T + p T^{2} ) \)
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

−8.007555618428826104242728859583, −7.49850214773930224745669921186, −7.14727326785100013983164955149, −6.68353609725862071234751274526, −6.27846377324631673216759255706, −5.80133561424136186244396129052, −5.20543613114974569546036540080, −4.96992360744408689484552152692, −4.43589661383561091247720235421, −3.75319460357054553986837961106, −3.24766601307650882030591436321, −3.06741153601361037751736769887, −2.16473739668213439074315963594, −1.40030972244161054669622621365, −0.54438636723677456574594616808, 0.54438636723677456574594616808, 1.40030972244161054669622621365, 2.16473739668213439074315963594, 3.06741153601361037751736769887, 3.24766601307650882030591436321, 3.75319460357054553986837961106, 4.43589661383561091247720235421, 4.96992360744408689484552152692, 5.20543613114974569546036540080, 5.80133561424136186244396129052, 6.27846377324631673216759255706, 6.68353609725862071234751274526, 7.14727326785100013983164955149, 7.49850214773930224745669921186, 8.007555618428826104242728859583

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