Properties

Label 4-930e2-1.1-c1e2-0-13
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 − 4·7-s + 4·8-s + 2·10-s + 3·11-s + 3·12-s + 2·13-s − 8·14-s + 15-s + 5·16-s − 3·17-s + 3·20-s − 4·21-s + 6·22-s + 6·23-s + 4·24-s + 4·26-s − 27-s − 12·28-s + 4·29-s + 2·30-s + 4·31-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 − 1.51·7-s + 1.41·8-s + 0.632·10-s + 0.904·11-s + 0.866·12-s + 0.554·13-s − 2.13·14-s + 0.258·15-s + 5/4·16-s − 0.727·17-s + 0.670·20-s − 0.872·21-s + 1.27·22-s + 1.25·23-s + 0.816·24-s + 0.784·26-s − 0.192·27-s − 2.26·28-s + 0.742·29-s + 0.365·30-s + 0.718·31-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: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 864900,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(6.555500076\)
\(L(\frac12)\) \(\approx\) \(6.555500076\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - 4 T + p T^{2} \)
good7$C_2$ \( ( 1 - T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
11$C_2^2$ \( 1 - 3 T - 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
17$C_2^2$ \( 1 + 3 T - 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
23$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 11 T + p T^{2} ) \)
41$C_2^2$ \( 1 - 2 T - 37 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
43$C_2^2$ \( 1 - T - 42 T^{2} - p T^{3} + p^{2} T^{4} \)
47$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
53$C_2^2$ \( 1 + 4 T - 37 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 4 T - 43 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
71$C_2^2$ \( 1 + 6 T - 35 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
73$C_2^2$ \( 1 - 14 T + 123 T^{2} - 14 p T^{3} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 17 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
83$C_2^2$ \( 1 + 12 T + 61 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \)
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

−10.09614873018416241204703557101, −10.09525906558232194306946797242, −9.303161367808781556979367266984, −9.199584065793971541431103152329, −8.569516861440567197812080845400, −8.341615241896789218493879833880, −7.42279795918684088903537232660, −7.22425382334881124565689590245, −6.49775466888027641132114124234, −6.43982957829245950124572615205, −6.15496752736418220895976880189, −5.47668710373780380993700138658, −4.93861677558081285453540997981, −4.48950715216738713579792303945, −3.72833159292597274449965840144, −3.66446007861820986766826500574, −2.86405044815947304372246796157, −2.69008970981109171078380846317, −1.86724401278316700695794125397, −0.968926216923291339214874450952, 0.968926216923291339214874450952, 1.86724401278316700695794125397, 2.69008970981109171078380846317, 2.86405044815947304372246796157, 3.66446007861820986766826500574, 3.72833159292597274449965840144, 4.48950715216738713579792303945, 4.93861677558081285453540997981, 5.47668710373780380993700138658, 6.15496752736418220895976880189, 6.43982957829245950124572615205, 6.49775466888027641132114124234, 7.22425382334881124565689590245, 7.42279795918684088903537232660, 8.341615241896789218493879833880, 8.569516861440567197812080845400, 9.199584065793971541431103152329, 9.303161367808781556979367266984, 10.09525906558232194306946797242, 10.09614873018416241204703557101

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