Properties

Degree $4$
Conductor $6615$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 4-s − 5-s + 9-s + 12-s − 15-s − 3·16-s − 4·17-s − 20-s − 2·25-s + 27-s + 36-s − 4·37-s + 12·41-s + 8·43-s − 45-s − 3·48-s − 7·49-s − 4·51-s − 8·59-s − 60-s − 7·64-s − 8·67-s − 4·68-s − 2·75-s + 3·80-s + 81-s + ⋯
L(s)  = 1  + 0.577·3-s + 1/2·4-s − 0.447·5-s + 1/3·9-s + 0.288·12-s − 0.258·15-s − 3/4·16-s − 0.970·17-s − 0.223·20-s − 2/5·25-s + 0.192·27-s + 1/6·36-s − 0.657·37-s + 1.87·41-s + 1.21·43-s − 0.149·45-s − 0.433·48-s − 49-s − 0.560·51-s − 1.04·59-s − 0.129·60-s − 7/8·64-s − 0.977·67-s − 0.485·68-s − 0.230·75-s + 0.335·80-s + 1/9·81-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: $1$
Motivic weight: \(1\)
Character: $\chi_{6615} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 6615,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.065470266\)
\(L(\frac12)\) \(\approx\) \(1.065470266\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
17$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
23$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
31$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \)
37$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
41$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 - 2 T + p T^{2} ) \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
61$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
67$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
71$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
73$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + p T^{2} )^{2} \)
83$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
89$C_2$$\times$$C_2$ \( ( 1 - 18 T + p T^{2} )( 1 - 10 T + p T^{2} ) \)
97$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \)
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

−11.96674570528888352784590256031, −11.28078562151763228684236334049, −10.90551042382422400651026208050, −10.36016771654528399053265104347, −9.335539872920611783638396169880, −9.199866042579042052517471539219, −8.382212856146157064902519405280, −7.72009981120177449248701273285, −7.22495947379405190721708338248, −6.52575558803626067260081988600, −5.84891730358471501690019697703, −4.68155243915535530135391274133, −4.11619187207847055059962499072, −3.03094143106400761993417036804, −2.07915336094385051156894345324, 2.07915336094385051156894345324, 3.03094143106400761993417036804, 4.11619187207847055059962499072, 4.68155243915535530135391274133, 5.84891730358471501690019697703, 6.52575558803626067260081988600, 7.22495947379405190721708338248, 7.72009981120177449248701273285, 8.382212856146157064902519405280, 9.199866042579042052517471539219, 9.335539872920611783638396169880, 10.36016771654528399053265104347, 10.90551042382422400651026208050, 11.28078562151763228684236334049, 11.96674570528888352784590256031

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