Properties

Label 2-43-1.1-c7-0-21
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  + 15.2·2-s − 9.20·3-s + 103.·4-s − 537.·5-s − 140.·6-s + 1.47e3·7-s − 371.·8-s − 2.10e3·9-s − 8.18e3·10-s − 3.35e3·11-s − 953.·12-s − 1.24e4·13-s + 2.23e4·14-s + 4.95e3·15-s − 1.89e4·16-s + 2.07e4·17-s − 3.19e4·18-s − 1.06e3·19-s − 5.57e4·20-s − 1.35e4·21-s − 5.10e4·22-s − 1.78e4·23-s + 3.41e3·24-s + 2.11e5·25-s − 1.88e5·26-s + 3.94e4·27-s + 1.52e5·28-s + ⋯
L(s)  = 1  + 1.34·2-s − 0.196·3-s + 0.809·4-s − 1.92·5-s − 0.264·6-s + 1.62·7-s − 0.256·8-s − 0.961·9-s − 2.58·10-s − 0.759·11-s − 0.159·12-s − 1.56·13-s + 2.18·14-s + 0.378·15-s − 1.15·16-s + 1.02·17-s − 1.29·18-s − 0.0356·19-s − 1.55·20-s − 0.319·21-s − 1.02·22-s − 0.305·23-s + 0.0504·24-s + 2.70·25-s − 2.10·26-s + 0.386·27-s + 1.31·28-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 \( 1 - 15.2T + 128T^{2} \)
3 \( 1 + 9.20T + 2.18e3T^{2} \)
5 \( 1 + 537.T + 7.81e4T^{2} \)
7 \( 1 - 1.47e3T + 8.23e5T^{2} \)
11 \( 1 + 3.35e3T + 1.94e7T^{2} \)
13 \( 1 + 1.24e4T + 6.27e7T^{2} \)
17 \( 1 - 2.07e4T + 4.10e8T^{2} \)
19 \( 1 + 1.06e3T + 8.93e8T^{2} \)
23 \( 1 + 1.78e4T + 3.40e9T^{2} \)
29 \( 1 - 9.99e4T + 1.72e10T^{2} \)
31 \( 1 + 3.01e4T + 2.75e10T^{2} \)
37 \( 1 + 1.74e5T + 9.49e10T^{2} \)
41 \( 1 + 4.50e4T + 1.94e11T^{2} \)
47 \( 1 + 3.19e5T + 5.06e11T^{2} \)
53 \( 1 - 7.27e4T + 1.17e12T^{2} \)
59 \( 1 + 2.67e6T + 2.48e12T^{2} \)
61 \( 1 + 2.37e6T + 3.14e12T^{2} \)
67 \( 1 - 3.70e6T + 6.06e12T^{2} \)
71 \( 1 + 4.04e6T + 9.09e12T^{2} \)
73 \( 1 - 8.62e5T + 1.10e13T^{2} \)
79 \( 1 + 3.18e6T + 1.92e13T^{2} \)
83 \( 1 + 2.05e6T + 2.71e13T^{2} \)
89 \( 1 + 1.05e7T + 4.42e13T^{2} \)
97 \( 1 - 8.49e5T + 8.07e13T^{2} \)
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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

−14.24829874917891296871293528803, −12.29139498529374763336053955895, −11.89687464252397826924366439319, −10.94320768679916717062558042709, −8.335404974707889501340124858260, −7.47764080909922140961518115645, −5.23970219426776687566954600673, −4.51014645997480542315599378891, −2.96438791237467688028337056439, 0, 2.96438791237467688028337056439, 4.51014645997480542315599378891, 5.23970219426776687566954600673, 7.47764080909922140961518115645, 8.335404974707889501340124858260, 10.94320768679916717062558042709, 11.89687464252397826924366439319, 12.29139498529374763336053955895, 14.24829874917891296871293528803

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