Properties

Label 2-19-1.1-c7-0-7
Degree $2$
Conductor $19$
Sign $-1$
Analytic cond. $5.93531$
Root an. cond. $2.43625$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4.43·2-s + 22.5·3-s − 108.·4-s + 160.·5-s − 99.9·6-s − 1.31e3·7-s + 1.04e3·8-s − 1.67e3·9-s − 711.·10-s − 3.32e3·11-s − 2.44e3·12-s − 1.43e4·13-s + 5.82e3·14-s + 3.61e3·15-s + 9.22e3·16-s + 3.25e4·17-s + 7.43e3·18-s + 6.85e3·19-s − 1.73e4·20-s − 2.96e4·21-s + 1.47e4·22-s + 5.61e4·23-s + 2.36e4·24-s − 5.23e4·25-s + 6.37e4·26-s − 8.71e4·27-s + 1.42e5·28-s + ⋯
L(s)  = 1  − 0.391·2-s + 0.482·3-s − 0.846·4-s + 0.574·5-s − 0.188·6-s − 1.44·7-s + 0.723·8-s − 0.767·9-s − 0.224·10-s − 0.753·11-s − 0.408·12-s − 1.81·13-s + 0.567·14-s + 0.276·15-s + 0.563·16-s + 1.60·17-s + 0.300·18-s + 0.229·19-s − 0.485·20-s − 0.698·21-s + 0.295·22-s + 0.961·23-s + 0.348·24-s − 0.670·25-s + 0.711·26-s − 0.852·27-s + 1.22·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(19\)
Sign: $-1$
Analytic conductor: \(5.93531\)
Root analytic conductor: \(2.43625\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 19,\ (\ :7/2),\ -1)\)

Particular Values

\(L(4)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad19 \( 1 - 6.85e3T \)
good2 \( 1 + 4.43T + 128T^{2} \)
3 \( 1 - 22.5T + 2.18e3T^{2} \)
5 \( 1 - 160.T + 7.81e4T^{2} \)
7 \( 1 + 1.31e3T + 8.23e5T^{2} \)
11 \( 1 + 3.32e3T + 1.94e7T^{2} \)
13 \( 1 + 1.43e4T + 6.27e7T^{2} \)
17 \( 1 - 3.25e4T + 4.10e8T^{2} \)
23 \( 1 - 5.61e4T + 3.40e9T^{2} \)
29 \( 1 + 1.50e5T + 1.72e10T^{2} \)
31 \( 1 - 1.88e5T + 2.75e10T^{2} \)
37 \( 1 + 2.40e5T + 9.49e10T^{2} \)
41 \( 1 - 5.13e5T + 1.94e11T^{2} \)
43 \( 1 - 2.31e5T + 2.71e11T^{2} \)
47 \( 1 + 1.27e6T + 5.06e11T^{2} \)
53 \( 1 - 3.27e5T + 1.17e12T^{2} \)
59 \( 1 + 8.98e5T + 2.48e12T^{2} \)
61 \( 1 + 7.24e5T + 3.14e12T^{2} \)
67 \( 1 + 3.85e5T + 6.06e12T^{2} \)
71 \( 1 + 6.62e5T + 9.09e12T^{2} \)
73 \( 1 + 2.78e6T + 1.10e13T^{2} \)
79 \( 1 - 7.31e5T + 1.92e13T^{2} \)
83 \( 1 - 3.58e6T + 2.71e13T^{2} \)
89 \( 1 + 4.54e6T + 4.42e13T^{2} \)
97 \( 1 + 1.57e7T + 8.07e13T^{2} \)
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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

−16.58516802609800612343661118858, −14.73500242065832066878933888863, −13.60416389839911323020978294055, −12.51382454827112294637115858466, −9.998942305959373118290961247891, −9.406087985482269555575881065000, −7.68700989944601172530915363324, −5.45443163196034056463253453140, −2.99292688189522424754034179389, 0, 2.99292688189522424754034179389, 5.45443163196034056463253453140, 7.68700989944601172530915363324, 9.406087985482269555575881065000, 9.998942305959373118290961247891, 12.51382454827112294637115858466, 13.60416389839911323020978294055, 14.73500242065832066878933888863, 16.58516802609800612343661118858

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