Properties

Degree $4$
Conductor $49787136$
Sign $1$
Motivic weight $1$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·5-s − 4·11-s − 12·17-s − 8·19-s − 4·23-s + 4·25-s + 8·37-s − 12·41-s + 8·47-s + 4·53-s + 16·55-s + 8·61-s − 4·71-s + 24·73-s − 16·79-s − 8·83-s + 48·85-s − 20·89-s + 32·95-s + 8·97-s − 4·101-s + 20·107-s + 16·109-s + 20·113-s + 16·115-s − 2·121-s + 12·125-s + ⋯
L(s)  = 1  − 1.78·5-s − 1.20·11-s − 2.91·17-s − 1.83·19-s − 0.834·23-s + 4/5·25-s + 1.31·37-s − 1.87·41-s + 1.16·47-s + 0.549·53-s + 2.15·55-s + 1.02·61-s − 0.474·71-s + 2.80·73-s − 1.80·79-s − 0.878·83-s + 5.20·85-s − 2.11·89-s + 3.28·95-s + 0.812·97-s − 0.398·101-s + 1.93·107-s + 1.53·109-s + 1.88·113-s + 1.49·115-s − 0.181·121-s + 1.07·125-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 49787136 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49787136 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(49787136\)    =    \(2^{8} \cdot 3^{4} \cdot 7^{4}\)
Sign: $1$
Motivic weight: \(1\)
Character: induced by $\chi_{7056} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 49787136,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4280333003\)
\(L(\frac12)\) \(\approx\) \(0.4280333003\)
\(L(\frac{3}{2})\) 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 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5$D_{4}$ \( 1 + 4 T + 12 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
13$C_2^2$ \( 1 + 8 T^{2} + p^{2} T^{4} \)
17$D_{4}$ \( 1 + 12 T + 4 p T^{2} + 12 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 + 8 T + 46 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
23$C_4$ \( 1 + 4 T + 42 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \)
31$C_2^2$ \( 1 + 54 T^{2} + p^{2} T^{4} \)
37$D_{4}$ \( 1 - 8 T + 58 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 + 12 T + 100 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
43$C_2^2$ \( 1 - 42 T^{2} + p^{2} T^{4} \)
47$D_{4}$ \( 1 - 8 T + 38 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 + 46 T^{2} + p^{2} T^{4} \)
61$D_{4}$ \( 1 - 8 T + 88 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + 4 T + 74 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 24 T + 272 T^{2} - 24 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 16 T + 190 T^{2} + 16 p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
89$D_{4}$ \( 1 + 20 T + 260 T^{2} + 20 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 8 T + 192 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.054090956165664499363798203797, −8.012437678770445184436892882848, −7.31114849236637548532476428154, −7.14078403119717017939989591176, −6.82020922478327883916631513480, −6.49727186149885248734054802671, −5.92027569394826114739053773209, −5.87611761026627310988348119191, −5.10413538442563797675751713808, −4.80559736765093845296889035846, −4.34110919017062850911418562921, −4.28302287724941254840806905527, −3.83543487038440203422024619340, −3.60527737697794632329466490542, −2.68571669290287719778029408696, −2.68537873460491479600591587458, −1.94528330907915967234986383116, −1.91134776435486199040532092949, −0.57442474647227776412574777998, −0.26697693438307821778195009369, 0.26697693438307821778195009369, 0.57442474647227776412574777998, 1.91134776435486199040532092949, 1.94528330907915967234986383116, 2.68537873460491479600591587458, 2.68571669290287719778029408696, 3.60527737697794632329466490542, 3.83543487038440203422024619340, 4.28302287724941254840806905527, 4.34110919017062850911418562921, 4.80559736765093845296889035846, 5.10413538442563797675751713808, 5.87611761026627310988348119191, 5.92027569394826114739053773209, 6.49727186149885248734054802671, 6.82020922478327883916631513480, 7.14078403119717017939989591176, 7.31114849236637548532476428154, 8.012437678770445184436892882848, 8.054090956165664499363798203797

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