↑

Properties

Label 4-82e2-1.1-c1e2-0-5
Degree $4$
Conductor $6724$
Sign $1$
Analytic cond. $0.428728$
Root an. cond. $0.809180$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 3·4-s − 4·7-s + 4·8-s − 4·9-s − 8·14-s + 5·16-s + 4·17-s − 8·18-s − 8·19-s + 8·23-s − 2·25-s − 12·28-s + 8·29-s − 8·31-s + 6·32-s + 8·34-s − 12·36-s − 16·38-s − 2·41-s + 8·43-s + 16·46-s − 4·47-s − 4·50-s + 24·53-s − 16·56-s + 16·58-s + ⋯
L(s)  = 1  + 1.41·2-s + 3/2·4-s − 1.51·7-s + 1.41·8-s − 4/3·9-s − 2.13·14-s + 5/4·16-s + 0.970·17-s − 1.88·18-s − 1.83·19-s + 1.66·23-s − 2/5·25-s − 2.26·28-s + 1.48·29-s − 1.43·31-s + 1.06·32-s + 1.37·34-s − 2·36-s − 2.59·38-s − 0.312·41-s + 1.21·43-s + 2.35·46-s − 0.583·47-s − 0.565·50-s + 3.29·53-s − 2.13·56-s + 2.10·58-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6724 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6724 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(6724\)    =    \(2^{2} \cdot 41^{2}\)
Sign: $1$
Analytic conductor: \(0.428728\)
Root analytic conductor: \(0.809180\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 6724,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.621103395\)
\(L(\frac12)\) \(\approx\) \(1.621103395\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{2} \)
41$C_1$ \( ( 1 + T )^{2} \)
good3$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \)
5$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
7$D_{4}$ \( 1 + 4 T + 16 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 + p T^{2} )^{2} \)
17$C_4$ \( 1 - 4 T + 6 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 + 8 T + 52 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 - 8 T + 54 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 - 8 T + 42 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 + 8 T + 70 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
37$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - 8 T + 70 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + 4 T + 48 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
59$D_{4}$ \( 1 + 8 T + 126 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
67$D_{4}$ \( 1 + 8 T + 132 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + 4 T + 144 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 16 T + 178 T^{2} + 16 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 12 T + 176 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 24 T + 278 T^{2} - 24 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 12 T + 182 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 4 T + 166 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−14.61501708324602010139292220637, −14.08919181679431614020214131365, −13.26830258684954251367887342932, −13.23854526427826292609453785409, −12.43006960413634175283749753821, −12.28054029136310314768439061782, −11.45708666641992929035560991919, −10.98613576177527262067132427299, −10.36072055623948630057110844562, −9.841007973617315997953823221716, −8.802986449848110905891376609478, −8.587271008673712693800179999139, −7.39832926665039599439095721645, −6.87569485664033031084846023607, −6.10668064570171486137898034335, −5.80970613597639718234982729359, −4.96800621436173310389501551718, −3.97049665509507487475106986467, −3.19054942816160639424782225438, −2.58987851377818515343201177613, 2.58987851377818515343201177613, 3.19054942816160639424782225438, 3.97049665509507487475106986467, 4.96800621436173310389501551718, 5.80970613597639718234982729359, 6.10668064570171486137898034335, 6.87569485664033031084846023607, 7.39832926665039599439095721645, 8.587271008673712693800179999139, 8.802986449848110905891376609478, 9.841007973617315997953823221716, 10.36072055623948630057110844562, 10.98613576177527262067132427299, 11.45708666641992929035560991919, 12.28054029136310314768439061782, 12.43006960413634175283749753821, 13.23854526427826292609453785409, 13.26830258684954251367887342932, 14.08919181679431614020214131365, 14.61501708324602010139292220637

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