Properties

Label 4-5390e2-1.1-c1e2-0-9
Degree $4$
Conductor $29052100$
Sign $1$
Analytic cond. $1852.38$
Root an. cond. $6.56043$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s − 4·3-s + 3·4-s − 2·5-s − 8·6-s + 4·8-s + 6·9-s − 4·10-s − 2·11-s − 12·12-s − 2·13-s + 8·15-s + 5·16-s + 2·17-s + 12·18-s + 2·19-s − 6·20-s − 4·22-s − 2·23-s − 16·24-s + 3·25-s − 4·26-s + 4·27-s + 6·29-s + 16·30-s + 2·31-s + 6·32-s + ⋯
L(s)  = 1  + 1.41·2-s − 2.30·3-s + 3/2·4-s − 0.894·5-s − 3.26·6-s + 1.41·8-s + 2·9-s − 1.26·10-s − 0.603·11-s − 3.46·12-s − 0.554·13-s + 2.06·15-s + 5/4·16-s + 0.485·17-s + 2.82·18-s + 0.458·19-s − 1.34·20-s − 0.852·22-s − 0.417·23-s − 3.26·24-s + 3/5·25-s − 0.784·26-s + 0.769·27-s + 1.11·29-s + 2.92·30-s + 0.359·31-s + 1.06·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 29052100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29052100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(29052100\)    =    \(2^{2} \cdot 5^{2} \cdot 7^{4} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(1852.38\)
Root analytic conductor: \(6.56043\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 29052100,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{2} \)
5$C_1$ \( ( 1 + T )^{2} \)
7 \( 1 \)
11$C_1$ \( ( 1 + T )^{2} \)
good3$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
13$D_{4}$ \( 1 + 2 T - 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
17$C_2^2$ \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 - 2 T + 6 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 + 2 T + 14 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 - 6 T + 34 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 - 2 T + 30 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 + 10 T + 66 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
47$D_{4}$ \( 1 - 2 T + 62 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 14 T + 122 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 - 6 T + 94 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 14 T + 138 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
67$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
73$D_{4}$ \( 1 + 10 T + 138 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - 2 T + 126 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
89$D_{4}$ \( 1 + 8 T + 62 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 22 T + 282 T^{2} - 22 p T^{3} + p^{2} T^{4} \)
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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

−7.59471965842305314091240875892, −7.59154161660677950999432548461, −7.10360435904038059373510380335, −6.80358753631289929375493111142, −6.24968581749317715353090197289, −6.14081216098754702097066194935, −5.73860531449093247886090532115, −5.51476217510531510689775585078, −5.03492386835604576413119207284, −4.81711499995931590210333777053, −4.38069511526472404604370890392, −4.35178860057308288997217693828, −3.45448280560995696007721410006, −3.28146268891980534067498469243, −2.61721525277059225564127436279, −2.48710889561785124217079560572, −1.24804633024969991018383795786, −1.20457748447372953967386812996, 0, 0, 1.20457748447372953967386812996, 1.24804633024969991018383795786, 2.48710889561785124217079560572, 2.61721525277059225564127436279, 3.28146268891980534067498469243, 3.45448280560995696007721410006, 4.35178860057308288997217693828, 4.38069511526472404604370890392, 4.81711499995931590210333777053, 5.03492386835604576413119207284, 5.51476217510531510689775585078, 5.73860531449093247886090532115, 6.14081216098754702097066194935, 6.24968581749317715353090197289, 6.80358753631289929375493111142, 7.10360435904038059373510380335, 7.59154161660677950999432548461, 7.59471965842305314091240875892

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