Properties

Label 4-7448e2-1.1-c1e2-0-8
Degree $4$
Conductor $55472704$
Sign $1$
Analytic cond. $3536.98$
Root an. cond. $7.71184$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 4·3-s + 2·5-s + 6·9-s − 3·11-s − 4·13-s + 8·15-s − 7·17-s + 2·19-s − 7·25-s − 4·27-s − 8·29-s + 8·31-s − 12·33-s + 4·37-s − 16·39-s − 6·41-s − 2·43-s + 12·45-s − 3·47-s − 28·51-s + 2·53-s − 6·55-s + 8·57-s + 8·59-s − 17·61-s − 8·65-s + 6·71-s + ⋯
L(s)  = 1  + 2.30·3-s + 0.894·5-s + 2·9-s − 0.904·11-s − 1.10·13-s + 2.06·15-s − 1.69·17-s + 0.458·19-s − 7/5·25-s − 0.769·27-s − 1.48·29-s + 1.43·31-s − 2.08·33-s + 0.657·37-s − 2.56·39-s − 0.937·41-s − 0.304·43-s + 1.78·45-s − 0.437·47-s − 3.92·51-s + 0.274·53-s − 0.809·55-s + 1.05·57-s + 1.04·59-s − 2.17·61-s − 0.992·65-s + 0.712·71-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 55472704 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55472704 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(55472704\)    =    \(2^{6} \cdot 7^{4} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(3536.98\)
Root analytic conductor: \(7.71184\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{7448} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 55472704,\ (\ :1/2, 1/2),\ 1)\)

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 \( 1 \)
7 \( 1 \)
19$C_1$ \( ( 1 - T )^{2} \)
good3$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
5$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
11$D_{4}$ \( 1 + 3 T + 10 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
17$C_2^2$ \( 1 + 7 T + 32 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - 11 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
41$D_{4}$ \( 1 + 6 T + 34 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
43$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
47$D_{4}$ \( 1 + 3 T + 82 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 - 2 T + 50 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
59$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
61$D_{4}$ \( 1 + 17 T + 180 T^{2} + 17 p T^{3} + p^{2} T^{4} \)
67$C_2$ \( ( 1 + p T^{2} )^{2} \)
71$D_{4}$ \( 1 - 6 T + 94 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 15 T + 188 T^{2} + 15 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 14 T + 150 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 8 T + 125 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 18 T + 202 T^{2} + 18 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 2 T + 138 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
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

−7.71714917823861638993109811044, −7.49511818147467074546082005126, −7.23104876612701806506903569807, −6.90859687520260318448077582540, −6.20283407147173623240099769954, −6.09249779661824202530871607201, −5.55528799533582143705735827223, −5.33064933673659869049511180988, −4.82300820846674955345054979549, −4.35780900733287766682040458910, −4.03452564750507730897093142648, −3.71506135383554409957625003259, −2.97466419594173721666056639922, −2.88681399415882883247649028216, −2.51642645950403880373160898772, −2.25742215999456096568911350087, −1.68847621078451017832376543411, −1.56299965826322635191903234855, 0, 0, 1.56299965826322635191903234855, 1.68847621078451017832376543411, 2.25742215999456096568911350087, 2.51642645950403880373160898772, 2.88681399415882883247649028216, 2.97466419594173721666056639922, 3.71506135383554409957625003259, 4.03452564750507730897093142648, 4.35780900733287766682040458910, 4.82300820846674955345054979549, 5.33064933673659869049511180988, 5.55528799533582143705735827223, 6.09249779661824202530871607201, 6.20283407147173623240099769954, 6.90859687520260318448077582540, 7.23104876612701806506903569807, 7.49511818147467074546082005126, 7.71714917823861638993109811044

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