Properties

Label 4-930e2-1.1-c1e2-0-10
Degree $4$
Conductor $864900$
Sign $1$
Analytic cond. $55.1467$
Root an. cond. $2.72508$
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 + 3-s + 3·4-s + 5-s − 2·6-s + 3·7-s − 4·8-s − 2·10-s + 3·11-s + 3·12-s − 4·13-s − 6·14-s + 15-s + 5·16-s + 4·17-s + 3·20-s + 3·21-s − 6·22-s + 4·23-s − 4·24-s + 8·26-s − 27-s + 9·28-s + 2·29-s − 2·30-s − 7·31-s − 6·32-s + ⋯
L(s)  = 1  − 1.41·2-s + 0.577·3-s + 3/2·4-s + 0.447·5-s − 0.816·6-s + 1.13·7-s − 1.41·8-s − 0.632·10-s + 0.904·11-s + 0.866·12-s − 1.10·13-s − 1.60·14-s + 0.258·15-s + 5/4·16-s + 0.970·17-s + 0.670·20-s + 0.654·21-s − 1.27·22-s + 0.834·23-s − 0.816·24-s + 1.56·26-s − 0.192·27-s + 1.70·28-s + 0.371·29-s − 0.365·30-s − 1.25·31-s − 1.06·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 864900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(864900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 31^{2}\)
Sign: $1$
Analytic conductor: \(55.1467\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{930} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 864900,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.787975875\)
\(L(\frac12)\) \(\approx\) \(1.787975875\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - T + T^{2} \)
5$C_2$ \( 1 - T + T^{2} \)
31$C_2$ \( 1 + 7 T + p T^{2} \)
good7$C_2^2$ \( 1 - 3 T + 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - 3 T - 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
13$C_2^2$ \( 1 + 4 T + 3 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
17$C_2^2$ \( 1 - 4 T - T^{2} - 4 p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
23$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 - 6 T - T^{2} - 6 p T^{3} + p^{2} T^{4} \)
41$C_2^2$ \( 1 - 2 T - 37 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
43$C_2^2$ \( 1 + 4 T - 27 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
47$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
53$C_2^2$ \( 1 - 3 T - 44 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 9 T + 22 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
61$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 + 2 T - 63 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
71$C_2^2$ \( 1 + 4 T - 55 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
73$C_2^2$ \( 1 + 2 T - 69 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 17 T + p T^{2} )( 1 + 13 T + p T^{2} ) \)
83$C_2^2$ \( 1 + 9 T - 2 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 11 T + p T^{2} )^{2} \)
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

−10.07798911326150405737174883869, −9.830290058217555949582316951872, −9.187963647352689207280972916550, −9.187294919342815267317648382655, −8.742711826524714251020359689481, −8.106166451005508725014107280902, −7.949518320989708776429499444316, −7.47204458621880329207051667070, −6.92710271944468010358143007498, −6.89837681649799638410317250595, −5.87566347793163294273825009038, −5.72655549257289422337288064574, −5.08904831039211667915392591074, −4.53356909219669297757417948322, −3.88979642042421673933644973386, −3.22078953128042079407789685374, −2.60373812176604309632083487874, −2.09682530844649868937651155686, −1.49264767003436741142285984151, −0.824254013008525873639844661898, 0.824254013008525873639844661898, 1.49264767003436741142285984151, 2.09682530844649868937651155686, 2.60373812176604309632083487874, 3.22078953128042079407789685374, 3.88979642042421673933644973386, 4.53356909219669297757417948322, 5.08904831039211667915392591074, 5.72655549257289422337288064574, 5.87566347793163294273825009038, 6.89837681649799638410317250595, 6.92710271944468010358143007498, 7.47204458621880329207051667070, 7.949518320989708776429499444316, 8.106166451005508725014107280902, 8.742711826524714251020359689481, 9.187294919342815267317648382655, 9.187963647352689207280972916550, 9.830290058217555949582316951872, 10.07798911326150405737174883869

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