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 − 8-s + 9-s − 10-s − 4·11-s − 12-s − 15-s + 16-s + 2·17-s − 18-s + 19-s + 20-s + 4·22-s + 4·23-s + 24-s + 25-s − 27-s + 6·29-s + 30-s − 4·31-s − 32-s + 4·33-s − 2·34-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s − 0.353·8-s + 1/3·9-s − 0.316·10-s − 1.20·11-s − 0.288·12-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 + 0.852·22-s + 0.834·23-s + 0.204·24-s + 1/5·25-s − 0.192·27-s + 1.11·29-s + 0.182·30-s − 0.718·31-s − 0.176·32-s + 0.696·33-s − 0.342·34-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 \( 1 + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−13.95205087618447, −13.46813618519218, −12.98786226250956, −12.58254219761687, −11.95554315773605, −11.53340299638727, −11.01269667397546, −10.34933174177454, −10.26475238046852, −9.743155866761600, −9.003842701058309, −8.682269578292141, −7.913649090876848, −7.634438583044534, −6.907017958536523, −6.579472230999048, −5.855488135074640, −5.305108755701275, −5.068513324274246, −4.241396593184949, −3.422941789384794, −2.780623888982601, −2.289369919978739, −1.430438477889926, −0.8369745588319242, 0, 0.8369745588319242, 1.430438477889926, 2.289369919978739, 2.780623888982601, 3.422941789384794, 4.241396593184949, 5.068513324274246, 5.305108755701275, 5.855488135074640, 6.579472230999048, 6.907017958536523, 7.634438583044534, 7.913649090876848, 8.682269578292141, 9.003842701058309, 9.743155866761600, 10.26475238046852, 10.34933174177454, 11.01269667397546, 11.53340299638727, 11.95554315773605, 12.58254219761687, 12.98786226250956, 13.46813618519218, 13.95205087618447

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