Properties

Label 4-7e4-1.1-c3e2-0-3
Degree $4$
Conductor $2401$
Sign $1$
Analytic cond. $8.35842$
Root an. cond. $1.70032$
Motivic weight $3$
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-s − 2·3-s + 8·4-s + 16·5-s − 2·6-s + 23·8-s + 27·9-s + 16·10-s + 8·11-s − 16·12-s − 56·13-s − 32·15-s + 23·16-s + 54·17-s + 27·18-s − 110·19-s + 128·20-s + 8·22-s − 48·23-s − 46·24-s + 125·25-s − 56·26-s − 154·27-s − 220·29-s − 32·30-s + 12·31-s + 184·32-s + ⋯
L(s)  = 1  + 0.353·2-s − 0.384·3-s + 4-s + 1.43·5-s − 0.136·6-s + 1.01·8-s + 9-s + 0.505·10-s + 0.219·11-s − 0.384·12-s − 1.19·13-s − 0.550·15-s + 0.359·16-s + 0.770·17-s + 0.353·18-s − 1.32·19-s + 1.43·20-s + 0.0775·22-s − 0.435·23-s − 0.391·24-s + 25-s − 0.422·26-s − 1.09·27-s − 1.40·29-s − 0.194·30-s + 0.0695·31-s + 1.01·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2401 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2401 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2401\)    =    \(7^{4}\)
Sign: $1$
Analytic conductor: \(8.35842\)
Root analytic conductor: \(1.70032\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 2401,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(2.805653448\)
\(L(\frac12)\) \(\approx\) \(2.805653448\)
\(L(\frac{5}{2})\) 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)$
bad7 \( 1 \)
good2$C_2^2$ \( 1 - T - 7 T^{2} - p^{3} T^{3} + p^{6} T^{4} \)
3$C_2^2$ \( 1 + 2 T - 23 T^{2} + 2 p^{3} T^{3} + p^{6} T^{4} \)
5$C_2^2$ \( 1 - 16 T + 131 T^{2} - 16 p^{3} T^{3} + p^{6} T^{4} \)
11$C_2^2$ \( 1 - 8 T - 1267 T^{2} - 8 p^{3} T^{3} + p^{6} T^{4} \)
13$C_2$ \( ( 1 + 28 T + p^{3} T^{2} )^{2} \)
17$C_2^2$ \( 1 - 54 T - 1997 T^{2} - 54 p^{3} T^{3} + p^{6} T^{4} \)
19$C_2^2$ \( 1 + 110 T + 5241 T^{2} + 110 p^{3} T^{3} + p^{6} T^{4} \)
23$C_2^2$ \( 1 + 48 T - 9863 T^{2} + 48 p^{3} T^{3} + p^{6} T^{4} \)
29$C_2$ \( ( 1 + 110 T + p^{3} T^{2} )^{2} \)
31$C_2^2$ \( 1 - 12 T - 29647 T^{2} - 12 p^{3} T^{3} + p^{6} T^{4} \)
37$C_2^2$ \( 1 - 246 T + 9863 T^{2} - 246 p^{3} T^{3} + p^{6} T^{4} \)
41$C_2$ \( ( 1 + 182 T + p^{3} T^{2} )^{2} \)
43$C_2$ \( ( 1 - 128 T + p^{3} T^{2} )^{2} \)
47$C_2^2$ \( 1 - 324 T + 1153 T^{2} - 324 p^{3} T^{3} + p^{6} T^{4} \)
53$C_2^2$ \( 1 - 162 T - 122633 T^{2} - 162 p^{3} T^{3} + p^{6} T^{4} \)
59$C_2^2$ \( 1 - 810 T + 450721 T^{2} - 810 p^{3} T^{3} + p^{6} T^{4} \)
61$C_2^2$ \( 1 + 8 p T + 3 p^{2} T^{2} + 8 p^{4} T^{3} + p^{6} T^{4} \)
67$C_2^2$ \( 1 + 244 T - 241227 T^{2} + 244 p^{3} T^{3} + p^{6} T^{4} \)
71$C_2$ \( ( 1 + 768 T + p^{3} T^{2} )^{2} \)
73$C_2^2$ \( 1 + 702 T + 103787 T^{2} + 702 p^{3} T^{3} + p^{6} T^{4} \)
79$C_2^2$ \( 1 + 440 T - 299439 T^{2} + 440 p^{3} T^{3} + p^{6} T^{4} \)
83$C_2$ \( ( 1 - 1302 T + p^{3} T^{2} )^{2} \)
89$C_2^2$ \( 1 - 730 T - 172069 T^{2} - 730 p^{3} T^{3} + p^{6} T^{4} \)
97$C_2$ \( ( 1 + 294 T + p^{3} T^{2} )^{2} \)
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

−15.12449547211677895848558907652, −14.95928764465242081501564011250, −14.32159472098490108690740481478, −13.36103545347305961138218382611, −13.34551402561014419930281042466, −12.54759605560259208078556604664, −11.98062440207555109203400991660, −11.34566488586736534784652419080, −10.51231436876016165329947925052, −10.20260110566653581829369896777, −9.678773948849315475126704849014, −8.883811470031736088011551101749, −7.48518103583452484772492699557, −7.36064990897455679231355987419, −6.33374777178742990390553892608, −5.88548694663386714316467601804, −4.96186071834753755412435917722, −4.07313329170699156517493870062, −2.38851112186511548991445905713, −1.70256716586223987954185963424, 1.70256716586223987954185963424, 2.38851112186511548991445905713, 4.07313329170699156517493870062, 4.96186071834753755412435917722, 5.88548694663386714316467601804, 6.33374777178742990390553892608, 7.36064990897455679231355987419, 7.48518103583452484772492699557, 8.883811470031736088011551101749, 9.678773948849315475126704849014, 10.20260110566653581829369896777, 10.51231436876016165329947925052, 11.34566488586736534784652419080, 11.98062440207555109203400991660, 12.54759605560259208078556604664, 13.34551402561014419930281042466, 13.36103545347305961138218382611, 14.32159472098490108690740481478, 14.95928764465242081501564011250, 15.12449547211677895848558907652

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