Properties

Label 4-3234e2-1.1-c1e2-0-2
Degree $4$
Conductor $10458756$
Sign $1$
Analytic cond. $666.859$
Root an. cond. $5.08169$
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  − 2·2-s + 2·3-s + 3·4-s − 4·6-s − 4·8-s + 3·9-s + 2·11-s + 6·12-s + 8·13-s + 5·16-s + 6·17-s − 6·18-s − 4·22-s − 8·24-s − 3·25-s − 16·26-s + 4·27-s + 4·29-s + 8·31-s − 6·32-s + 4·33-s − 12·34-s + 9·36-s − 8·37-s + 16·39-s − 18·41-s + 8·43-s + ⋯
L(s)  = 1  − 1.41·2-s + 1.15·3-s + 3/2·4-s − 1.63·6-s − 1.41·8-s + 9-s + 0.603·11-s + 1.73·12-s + 2.21·13-s + 5/4·16-s + 1.45·17-s − 1.41·18-s − 0.852·22-s − 1.63·24-s − 3/5·25-s − 3.13·26-s + 0.769·27-s + 0.742·29-s + 1.43·31-s − 1.06·32-s + 0.696·33-s − 2.05·34-s + 3/2·36-s − 1.31·37-s + 2.56·39-s − 2.81·41-s + 1.21·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 10458756 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10458756 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(10458756\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{4} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(666.859\)
Root analytic conductor: \(5.08169\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{3234} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 10458756,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.217111046\)
\(L(\frac12)\) \(\approx\) \(3.217111046\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{2} \)
3$C_1$ \( ( 1 - T )^{2} \)
7 \( 1 \)
11$C_1$ \( ( 1 - T )^{2} \)
good5$C_2^2$ \( 1 + 3 T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
19$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
23$C_2^2$ \( 1 + 39 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
37$D_{4}$ \( 1 + 8 T + 62 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \)
43$D_{4}$ \( 1 - 8 T + 74 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + 8 T + p T^{2} + 8 p T^{3} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
59$D_{4}$ \( 1 - 8 T + 22 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 8 T + 75 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + 6 T + 31 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + 16 T + 178 T^{2} + 16 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 20 T + 218 T^{2} - 20 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - 16 T + 215 T^{2} - 16 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 10 T + 79 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 16 T + 214 T^{2} + 16 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 2 T + 83 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
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

−8.696189294109841831527476580199, −8.470235918383574025210388645234, −8.169971854920359980332266304665, −8.119481213758977546403198644653, −7.43561891862866588715401998641, −7.16554146366174752870743327077, −6.59172544213212610393390989694, −6.55660002192121550067105612549, −5.88194655561563126909706037517, −5.73323795139435005933552708828, −5.01660986466511968432386910326, −4.50655410260769816386160914600, −3.83178771513609563236988613981, −3.60081397328304825178492172226, −3.10975125577073049147902862156, −2.97226777859884400697404589723, −1.92624059830205828812394529498, −1.79510108368002855792555073389, −1.15446190102172941289420198828, −0.74548602570690916536978104487, 0.74548602570690916536978104487, 1.15446190102172941289420198828, 1.79510108368002855792555073389, 1.92624059830205828812394529498, 2.97226777859884400697404589723, 3.10975125577073049147902862156, 3.60081397328304825178492172226, 3.83178771513609563236988613981, 4.50655410260769816386160914600, 5.01660986466511968432386910326, 5.73323795139435005933552708828, 5.88194655561563126909706037517, 6.55660002192121550067105612549, 6.59172544213212610393390989694, 7.16554146366174752870743327077, 7.43561891862866588715401998641, 8.119481213758977546403198644653, 8.169971854920359980332266304665, 8.470235918383574025210388645234, 8.696189294109841831527476580199

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