Properties

Degree $2$
Conductor $96330$
Sign $-1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 3-s + 4-s − 5-s − 6-s − 4·7-s − 8-s + 9-s + 10-s + 4·11-s + 12-s + 4·14-s − 15-s + 16-s − 2·17-s − 18-s + 19-s − 20-s − 4·21-s − 4·22-s − 8·23-s − 24-s + 25-s + 27-s − 4·28-s + 6·29-s + 30-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s − 1.51·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s + 1.20·11-s + 0.288·12-s + 1.06·14-s − 0.258·15-s + 1/4·16-s − 0.485·17-s − 0.235·18-s + 0.229·19-s − 0.223·20-s − 0.872·21-s − 0.852·22-s − 1.66·23-s − 0.204·24-s + 1/5·25-s + 0.192·27-s − 0.755·28-s + 1.11·29-s + 0.182·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 96330 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 96330 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(96330\)    =    \(2 \cdot 3 \cdot 5 \cdot 13^{2} \cdot 19\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{96330} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 96330,\ (\ :1/2),\ -1)\)

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$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 - T \)
5 \( 1 + T \)
13 \( 1 \)
19 \( 1 - T \)
good7 \( 1 + 4 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - 12 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 12 T + p T^{2} \)
83 \( 1 - 8 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 18 T + p T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−14.04083107480217, −13.67300867331484, −12.91095153685034, −12.53407536007658, −12.12461189308092, −11.60432662870472, −11.03729502133724, −10.44191488689873, −9.909662418609226, −9.498875089282893, −9.138025416684333, −8.715793544520588, −7.991694722648688, −7.641286164975593, −6.974462878683492, −6.521387991729089, −6.143592576937955, −5.575817013776092, −4.419215392902881, −3.992932424796973, −3.643004710650981, −2.712252691759424, −2.539291897642103, −1.511591905109301, −0.8080692763726666, 0, 0.8080692763726666, 1.511591905109301, 2.539291897642103, 2.712252691759424, 3.643004710650981, 3.992932424796973, 4.419215392902881, 5.575817013776092, 6.143592576937955, 6.521387991729089, 6.974462878683492, 7.641286164975593, 7.991694722648688, 8.715793544520588, 9.138025416684333, 9.498875089282893, 9.909662418609226, 10.44191488689873, 11.03729502133724, 11.60432662870472, 12.12461189308092, 12.53407536007658, 12.91095153685034, 13.67300867331484, 14.04083107480217

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