Properties

Label 2-43-1.1-c7-0-6
Degree $2$
Conductor $43$
Sign $-1$
Analytic cond. $13.4325$
Root an. cond. $3.66504$
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  − 13.4·2-s − 87.0·3-s + 53.3·4-s + 131.·5-s + 1.17e3·6-s − 712.·7-s + 1.00e3·8-s + 5.39e3·9-s − 1.76e3·10-s + 6.10e3·11-s − 4.64e3·12-s − 7.92e3·13-s + 9.59e3·14-s − 1.14e4·15-s − 2.03e4·16-s − 1.63e4·17-s − 7.26e4·18-s + 3.95e4·19-s + 6.99e3·20-s + 6.20e4·21-s − 8.21e4·22-s + 2.92e4·23-s − 8.75e4·24-s − 6.09e4·25-s + 1.06e5·26-s − 2.79e5·27-s − 3.80e4·28-s + ⋯
L(s)  = 1  − 1.19·2-s − 1.86·3-s + 0.416·4-s + 0.469·5-s + 2.21·6-s − 0.784·7-s + 0.694·8-s + 2.46·9-s − 0.558·10-s + 1.38·11-s − 0.776·12-s − 1.00·13-s + 0.934·14-s − 0.873·15-s − 1.24·16-s − 0.806·17-s − 2.93·18-s + 1.32·19-s + 0.195·20-s + 1.46·21-s − 1.64·22-s + 0.502·23-s − 1.29·24-s − 0.779·25-s + 1.19·26-s − 2.72·27-s − 0.327·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(43\)
Sign: $-1$
Analytic conductor: \(13.4325\)
Root analytic conductor: \(3.66504\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{43} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 43,\ (\ :7/2),\ -1)\)

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)$
bad43 \( 1 - 7.95e4T \)
good2 \( 1 + 13.4T + 128T^{2} \)
3 \( 1 + 87.0T + 2.18e3T^{2} \)
5 \( 1 - 131.T + 7.81e4T^{2} \)
7 \( 1 + 712.T + 8.23e5T^{2} \)
11 \( 1 - 6.10e3T + 1.94e7T^{2} \)
13 \( 1 + 7.92e3T + 6.27e7T^{2} \)
17 \( 1 + 1.63e4T + 4.10e8T^{2} \)
19 \( 1 - 3.95e4T + 8.93e8T^{2} \)
23 \( 1 - 2.92e4T + 3.40e9T^{2} \)
29 \( 1 - 7.11e4T + 1.72e10T^{2} \)
31 \( 1 - 1.07e5T + 2.75e10T^{2} \)
37 \( 1 - 3.86e5T + 9.49e10T^{2} \)
41 \( 1 + 7.68e5T + 1.94e11T^{2} \)
47 \( 1 - 9.29e5T + 5.06e11T^{2} \)
53 \( 1 + 1.50e6T + 1.17e12T^{2} \)
59 \( 1 + 1.77e6T + 2.48e12T^{2} \)
61 \( 1 + 1.22e6T + 3.14e12T^{2} \)
67 \( 1 + 8.31e5T + 6.06e12T^{2} \)
71 \( 1 + 2.10e4T + 9.09e12T^{2} \)
73 \( 1 + 6.31e5T + 1.10e13T^{2} \)
79 \( 1 + 3.76e6T + 1.92e13T^{2} \)
83 \( 1 - 1.71e6T + 2.71e13T^{2} \)
89 \( 1 + 6.08e6T + 4.42e13T^{2} \)
97 \( 1 + 7.22e6T + 8.07e13T^{2} \)
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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

−13.61857808526654266706698422318, −12.23045937234941381291613579379, −11.28329494506955011585615621564, −9.992379285979658988448159923362, −9.422892785887211889315585876188, −7.19280931221569374200915913394, −6.23772209148749328684189776479, −4.65801383280346497820947174918, −1.26161444862495668862143372586, 0, 1.26161444862495668862143372586, 4.65801383280346497820947174918, 6.23772209148749328684189776479, 7.19280931221569374200915913394, 9.422892785887211889315585876188, 9.992379285979658988448159923362, 11.28329494506955011585615621564, 12.23045937234941381291613579379, 13.61857808526654266706698422318

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