Properties

Label 4-460e2-1.1-c1e2-0-2
Degree $4$
Conductor $211600$
Sign $1$
Analytic cond. $13.4917$
Root an. cond. $1.91653$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 2·5-s + 7-s − 9-s + 4·11-s − 3·13-s + 2·15-s + 17-s + 12·19-s + 21-s + 2·23-s + 3·25-s + 4·29-s − 2·31-s + 4·33-s + 2·35-s − 5·37-s − 3·39-s − 2·45-s − 3·47-s − 9·49-s + 51-s − 5·53-s + 8·55-s + 12·57-s + 5·59-s − 6·61-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.894·5-s + 0.377·7-s − 1/3·9-s + 1.20·11-s − 0.832·13-s + 0.516·15-s + 0.242·17-s + 2.75·19-s + 0.218·21-s + 0.417·23-s + 3/5·25-s + 0.742·29-s − 0.359·31-s + 0.696·33-s + 0.338·35-s − 0.821·37-s − 0.480·39-s − 0.298·45-s − 0.437·47-s − 9/7·49-s + 0.140·51-s − 0.686·53-s + 1.07·55-s + 1.58·57-s + 0.650·59-s − 0.768·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 211600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 211600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(211600\)    =    \(2^{4} \cdot 5^{2} \cdot 23^{2}\)
Sign: $1$
Analytic conductor: \(13.4917\)
Root analytic conductor: \(1.91653\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{460} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 211600,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.872589725\)
\(L(\frac12)\) \(\approx\) \(2.872589725\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5$C_1$ \( ( 1 - T )^{2} \)
23$C_1$ \( ( 1 - T )^{2} \)
good3$D_{4}$ \( 1 - T + 2 T^{2} - p T^{3} + p^{2} T^{4} \)
7$D_{4}$ \( 1 - T + 10 T^{2} - p T^{3} + p^{2} T^{4} \)
11$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
13$D_{4}$ \( 1 + 3 T + 24 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 - T + 30 T^{2} - p T^{3} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
29$D_{4}$ \( 1 - 4 T + 45 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 + 2 T - 5 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 + 5 T + 76 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
41$C_2^2$ \( 1 + 65 T^{2} + p^{2} T^{4} \)
43$C_2$ \( ( 1 + p T^{2} )^{2} \)
47$D_{4}$ \( 1 + 3 T + 58 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 5 T + 108 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 - 5 T + 86 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 6 T + 114 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + 13 T + 172 T^{2} + 13 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 - 12 T + 161 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 13 T + 184 T^{2} + 13 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - 18 T + 222 T^{2} - 18 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - T - 42 T^{2} - p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 - 12 T + 146 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 18 T + 258 T^{2} + 18 p T^{3} + p^{2} T^{4} \)
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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

−11.15705410614823363849934995922, −10.94314736612685034449771762042, −10.26069663919525500049489331790, −9.674658730492202812419269314646, −9.413567443793721788312754260038, −9.374044152979274899586995087479, −8.501210706086085441995353827281, −8.296926865390312795284774797651, −7.48780060020705545327932624574, −7.30908675113184739361458820548, −6.61546309066113470560991071299, −6.24974283601122405255968479225, −5.32955652034778769588441946677, −5.30024816944262028073283923701, −4.64545925771980260631492864359, −3.80906719852345292629112267819, −3.04746029768935324374883113211, −2.87681778251835371546309168946, −1.75356094746200088213432563723, −1.19389131664706820276214767776, 1.19389131664706820276214767776, 1.75356094746200088213432563723, 2.87681778251835371546309168946, 3.04746029768935324374883113211, 3.80906719852345292629112267819, 4.64545925771980260631492864359, 5.30024816944262028073283923701, 5.32955652034778769588441946677, 6.24974283601122405255968479225, 6.61546309066113470560991071299, 7.30908675113184739361458820548, 7.48780060020705545327932624574, 8.296926865390312795284774797651, 8.501210706086085441995353827281, 9.374044152979274899586995087479, 9.413567443793721788312754260038, 9.674658730492202812419269314646, 10.26069663919525500049489331790, 10.94314736612685034449771762042, 11.15705410614823363849934995922

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