Properties

Label 4-6615-1.1-c1e2-0-0
Degree $4$
Conductor $6615$
Sign $1$
Analytic cond. $0.421778$
Root an. cond. $0.805881$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 4-s − 5-s + 9-s + 2·11-s − 12-s + 4·13-s + 15-s − 3·16-s + 4·17-s + 6·19-s − 20-s − 4·23-s − 2·25-s − 27-s − 10·29-s + 2·31-s − 2·33-s + 36-s − 4·39-s − 4·41-s − 12·43-s + 2·44-s − 45-s − 8·47-s + 3·48-s − 7·49-s + ⋯
L(s)  = 1  − 0.577·3-s + 1/2·4-s − 0.447·5-s + 1/3·9-s + 0.603·11-s − 0.288·12-s + 1.10·13-s + 0.258·15-s − 3/4·16-s + 0.970·17-s + 1.37·19-s − 0.223·20-s − 0.834·23-s − 2/5·25-s − 0.192·27-s − 1.85·29-s + 0.359·31-s − 0.348·33-s + 1/6·36-s − 0.640·39-s − 0.624·41-s − 1.82·43-s + 0.301·44-s − 0.149·45-s − 1.16·47-s + 0.433·48-s − 49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6615 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6615 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(6615\)    =    \(3^{3} \cdot 5 \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(0.421778\)
Root analytic conductor: \(0.805881\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{6615} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 6615,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8295318623\)
\(L(\frac12)\) \(\approx\) \(0.8295318623\)
\(L(\frac{3}{2})\) 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$C_1$ \( 1 + T \)
5$C_1$$\times$$C_2$ \( ( 1 - T )( 1 + 2 T + p T^{2} ) \)
7$C_2$ \( 1 + p T^{2} \)
good2$C_2^2$ \( 1 - T^{2} + p^{2} T^{4} \)
11$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
13$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
17$C_4$ \( 1 - 4 T + 6 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 - 6 T + 30 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 + 4 T - 2 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
29$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
31$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
41$D_{4}$ \( 1 + 4 T - 26 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
43$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \)
47$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \)
53$D_{4}$ \( 1 - 8 T + 102 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 - 16 T + 134 T^{2} - 16 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 2 T - 70 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 - 74 T^{2} + p^{2} T^{4} \)
71$D_{4}$ \( 1 - 2 T - 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
73$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
79$D_{4}$ \( 1 - 8 T + 126 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
89$C_2$$\times$$C_2$ \( ( 1 - 18 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
97$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
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

−16.9208924383, −16.4630874841, −16.2397566666, −15.7964294299, −15.1502143415, −14.7468425365, −14.0699434937, −13.3660582775, −13.2200131717, −12.2105321664, −11.7512559818, −11.4769745016, −11.1194206883, −10.1374491769, −9.83983589356, −9.07297702439, −8.30842730464, −7.71581474038, −7.03712971143, −6.45370608558, −5.73222474399, −5.08259781538, −3.92784521913, −3.37169891372, −1.66658984670, 1.66658984670, 3.37169891372, 3.92784521913, 5.08259781538, 5.73222474399, 6.45370608558, 7.03712971143, 7.71581474038, 8.30842730464, 9.07297702439, 9.83983589356, 10.1374491769, 11.1194206883, 11.4769745016, 11.7512559818, 12.2105321664, 13.2200131717, 13.3660582775, 14.0699434937, 14.7468425365, 15.1502143415, 15.7964294299, 16.2397566666, 16.4630874841, 16.9208924383

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