Properties

Label 4-261e2-1.1-c1e2-0-1
Degree $4$
Conductor $68121$
Sign $1$
Analytic cond. $4.34345$
Root an. cond. $1.44363$
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 + 4-s + 2·5-s + 4·10-s − 2·11-s − 2·13-s + 16-s + 4·17-s + 12·19-s + 2·20-s − 4·22-s + 4·23-s − 7·25-s − 4·26-s − 2·29-s + 6·31-s − 2·32-s + 8·34-s − 8·37-s + 24·38-s − 8·41-s + 10·43-s − 2·44-s + 8·46-s − 2·47-s − 6·49-s − 14·50-s + ⋯
L(s)  = 1  + 1.41·2-s + 1/2·4-s + 0.894·5-s + 1.26·10-s − 0.603·11-s − 0.554·13-s + 1/4·16-s + 0.970·17-s + 2.75·19-s + 0.447·20-s − 0.852·22-s + 0.834·23-s − 7/5·25-s − 0.784·26-s − 0.371·29-s + 1.07·31-s − 0.353·32-s + 1.37·34-s − 1.31·37-s + 3.89·38-s − 1.24·41-s + 1.52·43-s − 0.301·44-s + 1.17·46-s − 0.291·47-s − 6/7·49-s − 1.97·50-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 68121 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 68121 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(68121\)    =    \(3^{4} \cdot 29^{2}\)
Sign: $1$
Analytic conductor: \(4.34345\)
Root analytic conductor: \(1.44363\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 68121,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.205408657\)
\(L(\frac12)\) \(\approx\) \(3.205408657\)
\(L(\frac{3}{2})\) 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)$
bad3 \( 1 \)
29$C_1$ \( ( 1 + T )^{2} \)
good2$D_{4}$ \( 1 - p T + 3 T^{2} - p^{2} T^{3} + p^{2} T^{4} \)
5$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
7$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \)
11$D_{4}$ \( 1 + 2 T + 21 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 + 2 T + 19 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
23$D_{4}$ \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 - 6 T + 21 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
37$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
41$D_{4}$ \( 1 + 8 T + 26 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - 10 T + 109 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + 2 T + 77 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 2 T + 35 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 4 T + 90 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 4 T + 118 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 102 T^{2} + p^{2} T^{4} \)
71$C_4$ \( 1 - 12 T + 170 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
73$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
79$D_{4}$ \( 1 + 2 T + 157 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 4 T + 138 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 - 8 T + 122 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 8 T + 138 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
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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

−12.23621555334175947924322218029, −11.93569657781104304588346859325, −11.53243840575287084719496578146, −10.88441069073864566278578553172, −10.13849306395019046040820746828, −10.00259838211924625211079851424, −9.430617703393691399289608024241, −9.109959952993404521557456510295, −8.144287798231186127049924787138, −7.61443142486533455472104152675, −7.38181838265147786855543681916, −6.56721286853034092137906157602, −5.79406226310636244204274697138, −5.51306925471979922895624780387, −5.00469368051621207263820822774, −4.74002701089451450913131013016, −3.50102159378500060775047755493, −3.42299608240273759632639367481, −2.44159806729669538613903024767, −1.38056529535526415642277269031, 1.38056529535526415642277269031, 2.44159806729669538613903024767, 3.42299608240273759632639367481, 3.50102159378500060775047755493, 4.74002701089451450913131013016, 5.00469368051621207263820822774, 5.51306925471979922895624780387, 5.79406226310636244204274697138, 6.56721286853034092137906157602, 7.38181838265147786855543681916, 7.61443142486533455472104152675, 8.144287798231186127049924787138, 9.109959952993404521557456510295, 9.430617703393691399289608024241, 10.00259838211924625211079851424, 10.13849306395019046040820746828, 10.88441069073864566278578553172, 11.53243840575287084719496578146, 11.93569657781104304588346859325, 12.23621555334175947924322218029

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