Properties

Label 4-1680e2-1.1-c1e2-0-7
Degree $4$
Conductor $2822400$
Sign $1$
Analytic cond. $179.958$
Root an. cond. $3.66263$
Motivic weight $1$
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  + 4·5-s − 9-s − 8·11-s − 4·19-s + 11·25-s − 12·29-s − 12·31-s − 4·45-s − 49-s − 32·55-s + 8·59-s − 4·61-s + 16·71-s + 32·79-s + 81-s − 32·89-s − 16·95-s + 8·99-s + 16·101-s − 36·109-s + 26·121-s + 24·125-s + 127-s + 131-s + 137-s + 139-s − 48·145-s + ⋯
L(s)  = 1  + 1.78·5-s − 1/3·9-s − 2.41·11-s − 0.917·19-s + 11/5·25-s − 2.22·29-s − 2.15·31-s − 0.596·45-s − 1/7·49-s − 4.31·55-s + 1.04·59-s − 0.512·61-s + 1.89·71-s + 3.60·79-s + 1/9·81-s − 3.39·89-s − 1.64·95-s + 0.804·99-s + 1.59·101-s − 3.44·109-s + 2.36·121-s + 2.14·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 3.98·145-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2822400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2822400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2822400\)    =    \(2^{8} \cdot 3^{2} \cdot 5^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(179.958\)
Root analytic conductor: \(3.66263\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1680} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 2822400,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.403052917\)
\(L(\frac12)\) \(\approx\) \(1.403052917\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T^{2} \)
5$C_2$ \( 1 - 4 T + p T^{2} \)
7$C_2$ \( 1 + T^{2} \)
good11$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
13$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
17$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
19$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 78 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 102 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 16 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 + 90 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 16 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
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

−9.473085835099832135961826760589, −9.309383403710638666001786121605, −8.892317637351689119864639239854, −8.238221615209891668067861805353, −8.129618539359861725128170546132, −7.46076405834372518269636748419, −7.31389556620405462878996554263, −6.59041345150838864876647244373, −6.35471301963850703238367892455, −5.68340562035333934168063884294, −5.48754715163448808142478910701, −5.16279447382091566237105160711, −5.00808821292139965791964853524, −3.96093171659562900994942581441, −3.70456521129806073514636610767, −2.74431134969039036229484213345, −2.64614280749033305500312946816, −1.84652139555756762753248856884, −1.84559586549780823792609125113, −0.41002701862452815638068003388, 0.41002701862452815638068003388, 1.84559586549780823792609125113, 1.84652139555756762753248856884, 2.64614280749033305500312946816, 2.74431134969039036229484213345, 3.70456521129806073514636610767, 3.96093171659562900994942581441, 5.00808821292139965791964853524, 5.16279447382091566237105160711, 5.48754715163448808142478910701, 5.68340562035333934168063884294, 6.35471301963850703238367892455, 6.59041345150838864876647244373, 7.31389556620405462878996554263, 7.46076405834372518269636748419, 8.129618539359861725128170546132, 8.238221615209891668067861805353, 8.892317637351689119864639239854, 9.309383403710638666001786121605, 9.473085835099832135961826760589

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