Properties

Label 4-960e2-1.1-c2e2-0-0
Degree $4$
Conductor $921600$
Sign $1$
Analytic cond. $684.246$
Root an. cond. $5.11449$
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  + 4·3-s − 12·7-s + 7·9-s − 32·13-s + 4·19-s − 48·21-s − 5·25-s − 8·27-s − 36·31-s + 32·37-s − 128·39-s − 32·43-s + 10·49-s + 16·57-s − 164·61-s − 84·63-s − 48·67-s − 148·73-s − 20·75-s + 276·79-s − 95·81-s + 384·91-s − 144·93-s − 332·97-s + 52·103-s − 76·109-s + 128·111-s + ⋯
L(s)  = 1  + 4/3·3-s − 1.71·7-s + 7/9·9-s − 2.46·13-s + 4/19·19-s − 2.28·21-s − 1/5·25-s − 0.296·27-s − 1.16·31-s + 0.864·37-s − 3.28·39-s − 0.744·43-s + 0.204·49-s + 0.280·57-s − 2.68·61-s − 4/3·63-s − 0.716·67-s − 2.02·73-s − 0.266·75-s + 3.49·79-s − 1.17·81-s + 4.21·91-s − 1.54·93-s − 3.42·97-s + 0.504·103-s − 0.697·109-s + 1.15·111-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 921600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 921600 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(921600\)    =    \(2^{12} \cdot 3^{2} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(684.246\)
Root analytic conductor: \(5.11449\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{960} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 921600,\ (\ :1, 1),\ 1)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.1007212328\)
\(L(\frac12)\) \(\approx\) \(0.1007212328\)
\(L(2)\) 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 \( 1 \)
3$C_2$ \( 1 - 4 T + p^{2} T^{2} \)
5$C_2$ \( 1 + p T^{2} \)
good7$C_2$ \( ( 1 + 6 T + p^{2} T^{2} )^{2} \)
11$C_2^2$ \( 1 - 222 T^{2} + p^{4} T^{4} \)
13$C_2$ \( ( 1 + 16 T + p^{2} T^{2} )^{2} \)
17$C_2^2$ \( 1 - 558 T^{2} + p^{4} T^{4} \)
19$C_2$ \( ( 1 - 2 T + p^{2} T^{2} )^{2} \)
23$C_2$ \( ( 1 - 44 T + p^{2} T^{2} )( 1 + 44 T + p^{2} T^{2} ) \)
29$C_2^2$ \( 1 - 702 T^{2} + p^{4} T^{4} \)
31$C_2$ \( ( 1 + 18 T + p^{2} T^{2} )^{2} \)
37$C_2$ \( ( 1 - 16 T + p^{2} T^{2} )^{2} \)
41$C_2^2$ \( 1 + 558 T^{2} + p^{4} T^{4} \)
43$C_2$ \( ( 1 + 16 T + p^{2} T^{2} )^{2} \)
47$C_2^2$ \( 1 - 1998 T^{2} + p^{4} T^{4} \)
53$C_2^2$ \( 1 - 5598 T^{2} + p^{4} T^{4} \)
59$C_2^2$ \( 1 - 6942 T^{2} + p^{4} T^{4} \)
61$C_2$ \( ( 1 + 82 T + p^{2} T^{2} )^{2} \)
67$C_2$ \( ( 1 + 24 T + p^{2} T^{2} )^{2} \)
71$C_2^2$ \( 1 + 5598 T^{2} + p^{4} T^{4} \)
73$C_2$ \( ( 1 + 74 T + p^{2} T^{2} )^{2} \)
79$C_2$ \( ( 1 - 138 T + p^{2} T^{2} )^{2} \)
83$C_2^2$ \( 1 - 4958 T^{2} + p^{4} T^{4} \)
89$C_2$ \( ( 1 - 142 T + p^{2} T^{2} )( 1 + 142 T + p^{2} T^{2} ) \)
97$C_2$ \( ( 1 + 166 T + p^{2} T^{2} )^{2} \)
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

−10.10072915033844897428547874623, −9.632789661362157763880760345370, −9.369317987145357652346132520001, −8.909903037297951816079553736473, −8.453390269387523133897739370828, −7.75737669104397396143380662810, −7.64047378899354147729507971155, −7.15067921766866296835217335077, −6.87044966700301108406721892460, −6.05577318849746578906627147384, −6.02148426302105952664043032474, −4.93971733998390532127082058106, −4.93326657896860365665155192059, −4.05494594856040619999608653792, −3.60653301888819599684046253340, −3.00132364844718114503731334732, −2.79605874642251400436888695018, −2.25058790495281515450366470130, −1.53133951491367683391982936359, −0.086027512788810667266821994792, 0.086027512788810667266821994792, 1.53133951491367683391982936359, 2.25058790495281515450366470130, 2.79605874642251400436888695018, 3.00132364844718114503731334732, 3.60653301888819599684046253340, 4.05494594856040619999608653792, 4.93326657896860365665155192059, 4.93971733998390532127082058106, 6.02148426302105952664043032474, 6.05577318849746578906627147384, 6.87044966700301108406721892460, 7.15067921766866296835217335077, 7.64047378899354147729507971155, 7.75737669104397396143380662810, 8.453390269387523133897739370828, 8.909903037297951816079553736473, 9.369317987145357652346132520001, 9.632789661362157763880760345370, 10.10072915033844897428547874623

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