Properties

Label 4-930e2-1.1-c1e2-0-20
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 + 2·3-s + 3·4-s + 2·5-s + 4·6-s + 7-s + 4·8-s + 3·9-s + 4·10-s + 7·11-s + 6·12-s − 4·13-s + 2·14-s + 4·15-s + 5·16-s + 2·17-s + 6·18-s − 7·19-s + 6·20-s + 2·21-s + 14·22-s + 23-s + 8·24-s + 3·25-s − 8·26-s + 4·27-s + 3·28-s + ⋯
L(s)  = 1  + 1.41·2-s + 1.15·3-s + 3/2·4-s + 0.894·5-s + 1.63·6-s + 0.377·7-s + 1.41·8-s + 9-s + 1.26·10-s + 2.11·11-s + 1.73·12-s − 1.10·13-s + 0.534·14-s + 1.03·15-s + 5/4·16-s + 0.485·17-s + 1.41·18-s − 1.60·19-s + 1.34·20-s + 0.436·21-s + 2.98·22-s + 0.208·23-s + 1.63·24-s + 3/5·25-s − 1.56·26-s + 0.769·27-s + 0.566·28-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

\(L(1)\) \(\approx\) \(11.06306073\)
\(L(\frac12)\) \(\approx\) \(11.06306073\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{2} \)
5$C_1$ \( ( 1 - T )^{2} \)
31$C_1$ \( ( 1 + T )^{2} \)
good7$D_{4}$ \( 1 - T + 6 T^{2} - p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 - 7 T + 26 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
17$C_2^2$ \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 + 7 T + 42 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 - T + 38 T^{2} - p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 + 6 T + 34 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 - 10 T + 66 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 + 10 T + 74 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - 7 T + 90 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + 2 T + 62 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 3 T + 100 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 6 T + 94 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
67$D_{4}$ \( 1 - 10 T + 126 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 - T + 134 T^{2} - p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - T + 72 T^{2} - p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 15 T + 206 T^{2} + 15 p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
89$D_{4}$ \( 1 - 3 T + 172 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
97$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
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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

−10.10122555632368442110937507504, −9.845877171217146095031777213704, −9.456302010721688682340875744034, −9.151686987727392263319328945180, −8.432811589767721535782102862692, −8.383538037439911042697584044446, −7.45183342421001718272130718469, −7.35335038309971222476904616927, −6.70709419306710233180753915970, −6.43135534110010203914528061429, −5.96903423339873740311733271192, −5.48190214137245705985418789142, −4.76158486467898142834850524931, −4.52857870310880117502749956825, −3.85389128770315371773155464894, −3.71974408397570493107409839858, −2.79987541782877139287620011463, −2.50064394170691424181515377433, −1.68007154596548643645809445793, −1.49686357749362007261221879632, 1.49686357749362007261221879632, 1.68007154596548643645809445793, 2.50064394170691424181515377433, 2.79987541782877139287620011463, 3.71974408397570493107409839858, 3.85389128770315371773155464894, 4.52857870310880117502749956825, 4.76158486467898142834850524931, 5.48190214137245705985418789142, 5.96903423339873740311733271192, 6.43135534110010203914528061429, 6.70709419306710233180753915970, 7.35335038309971222476904616927, 7.45183342421001718272130718469, 8.383538037439911042697584044446, 8.432811589767721535782102862692, 9.151686987727392263319328945180, 9.456302010721688682340875744034, 9.845877171217146095031777213704, 10.10122555632368442110937507504

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