Properties

Label 4-930e2-1.1-c1e2-0-19
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

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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 + 2·13-s + 6·14-s + 15-s + 5·16-s + 4·17-s + 3·20-s + 3·21-s + 6·22-s − 8·23-s + 4·24-s + 4·26-s − 27-s + 9·28-s − 10·29-s + 2·30-s + 11·31-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 + 0.554·13-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 − 1.66·23-s + 0.816·24-s + 0.784·26-s − 0.192·27-s + 1.70·28-s − 1.85·29-s + 0.365·30-s + 1.97·31-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

\(L(1)\) \(\approx\) \(9.174095332\)
\(L(\frac12)\) \(\approx\) \(9.174095332\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - 11 T + p T^{2} \)
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$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
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$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
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$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
47$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
53$C_2^2$ \( 1 - 3 T - 44 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 + 3 T - 50 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
61$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 10 T + 33 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
71$C_2^2$ \( 1 - 8 T - 7 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
73$C_2^2$ \( 1 + 14 T + 123 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
79$C_2^2$ \( 1 + 8 T - 15 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
83$C_2^2$ \( 1 - 9 T - 2 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.15490639733422592398993750236, −10.06313336648752984205782750850, −9.291772934936483037937818557628, −9.259186178159370207996579597284, −8.355019401360742023099072822766, −7.976943537951356950966702021411, −7.929063340094521089099421073307, −7.31338212734660411489206356737, −6.69896880412838919509821790879, −6.26054626222910571322493439985, −5.75902762572781259140433119921, −5.68280636162624683313065548242, −4.86889098220833586022263647804, −4.50015076232325173498030338145, −3.87030646276061031301297979428, −3.76954057489232609354877021191, −2.87811219297513495013808418306, −2.46838012742712737135329819441, −1.62380532765996854711342145857, −1.36320197114878326546102890200, 1.36320197114878326546102890200, 1.62380532765996854711342145857, 2.46838012742712737135329819441, 2.87811219297513495013808418306, 3.76954057489232609354877021191, 3.87030646276061031301297979428, 4.50015076232325173498030338145, 4.86889098220833586022263647804, 5.68280636162624683313065548242, 5.75902762572781259140433119921, 6.26054626222910571322493439985, 6.69896880412838919509821790879, 7.31338212734660411489206356737, 7.929063340094521089099421073307, 7.976943537951356950966702021411, 8.355019401360742023099072822766, 9.259186178159370207996579597284, 9.291772934936483037937818557628, 10.06313336648752984205782750850, 10.15490639733422592398993750236

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