Properties

Label 4-6615-1.1-c1e2-0-3
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 3-s − 2·5-s + 6-s − 8-s − 2·9-s + 2·10-s − 11-s + 2·13-s + 2·15-s − 16-s − 4·17-s + 2·18-s − 4·19-s + 22-s − 6·23-s + 24-s + 2·25-s − 2·26-s + 5·27-s − 2·29-s − 2·30-s − 2·31-s + 6·32-s + 33-s + 4·34-s + 37-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s − 0.894·5-s + 0.408·6-s − 0.353·8-s − 2/3·9-s + 0.632·10-s − 0.301·11-s + 0.554·13-s + 0.516·15-s − 1/4·16-s − 0.970·17-s + 0.471·18-s − 0.917·19-s + 0.213·22-s − 1.25·23-s + 0.204·24-s + 2/5·25-s − 0.392·26-s + 0.962·27-s − 0.371·29-s − 0.365·30-s − 0.359·31-s + 1.06·32-s + 0.174·33-s + 0.685·34-s + 0.164·37-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: \(1\)
Selberg data: \((4,\ 6615,\ (\ :1/2, 1/2),\ -1)\)

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)$
bad3$C_2$ \( 1 + T + p T^{2} \)
5$C_1$$\times$$C_2$ \( ( 1 - T )( 1 + 3 T + p T^{2} ) \)
7$C_2$ \( 1 + p T^{2} \)
good2$D_{4}$ \( 1 + T + T^{2} + p T^{3} + p^{2} T^{4} \)
11$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
13$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
17$D_{4}$ \( 1 + 4 T + 25 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 + 4 T + 21 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 + 6 T + 31 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 + 2 T - 14 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
31$C_2$$\times$$C_2$ \( ( 1 - T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
37$D_{4}$ \( 1 - T - 24 T^{2} - p T^{3} + p^{2} T^{4} \)
41$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
43$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
47$D_{4}$ \( 1 - 5 T + 46 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 - 2 T + 67 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + T - 2 T^{2} + p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 - 2 T - 81 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 - 11 T + 68 T^{2} - 11 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + 2 T - 50 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + T + p T^{3} + p^{2} T^{4} \)
79$C_2$$\times$$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + T + p T^{2} ) \)
83$D_{4}$ \( 1 + 3 T + 82 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
89$C_2$$\times$$C_2$ \( ( 1 - 18 T + p T^{2} )( 1 + 15 T + p T^{2} ) \)
97$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−17.4462319024, −16.9614279132, −16.4018073440, −15.8861453200, −15.6183375834, −14.9427151821, −14.5085624307, −13.7691280962, −13.2480210384, −12.6290985143, −11.9965541194, −11.6067010190, −10.9950645755, −10.7679645797, −9.90482067937, −9.24296452798, −8.59273626610, −8.30639810479, −7.64206072380, −6.63922209059, −6.28013795230, −5.39862175636, −4.47629331790, −3.71826311639, −2.43573173983, 0, 2.43573173983, 3.71826311639, 4.47629331790, 5.39862175636, 6.28013795230, 6.63922209059, 7.64206072380, 8.30639810479, 8.59273626610, 9.24296452798, 9.90482067937, 10.7679645797, 10.9950645755, 11.6067010190, 11.9965541194, 12.6290985143, 13.2480210384, 13.7691280962, 14.5085624307, 14.9427151821, 15.6183375834, 15.8861453200, 16.4018073440, 16.9614279132, 17.4462319024

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