Properties

Label 2-43-1.1-c7-0-15
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  − 15.1·2-s + 62.8·3-s + 100.·4-s + 12.0·5-s − 951.·6-s − 1.24e3·7-s + 411.·8-s + 1.76e3·9-s − 182.·10-s − 2.04e3·11-s + 6.33e3·12-s + 9.46e3·13-s + 1.88e4·14-s + 758.·15-s − 1.91e4·16-s − 1.90e4·17-s − 2.67e4·18-s − 2.83e4·19-s + 1.21e3·20-s − 7.85e4·21-s + 3.09e4·22-s + 2.08e3·23-s + 2.58e4·24-s − 7.79e4·25-s − 1.43e5·26-s − 2.64e4·27-s − 1.25e5·28-s + ⋯
L(s)  = 1  − 1.33·2-s + 1.34·3-s + 0.787·4-s + 0.0431·5-s − 1.79·6-s − 1.37·7-s + 0.284·8-s + 0.807·9-s − 0.0576·10-s − 0.462·11-s + 1.05·12-s + 1.19·13-s + 1.83·14-s + 0.0580·15-s − 1.16·16-s − 0.939·17-s − 1.07·18-s − 0.947·19-s + 0.0339·20-s − 1.84·21-s + 0.618·22-s + 0.0357·23-s + 0.381·24-s − 0.998·25-s − 1.59·26-s − 0.258·27-s − 1.08·28-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 \( 1 + 15.1T + 128T^{2} \)
3 \( 1 - 62.8T + 2.18e3T^{2} \)
5 \( 1 - 12.0T + 7.81e4T^{2} \)
7 \( 1 + 1.24e3T + 8.23e5T^{2} \)
11 \( 1 + 2.04e3T + 1.94e7T^{2} \)
13 \( 1 - 9.46e3T + 6.27e7T^{2} \)
17 \( 1 + 1.90e4T + 4.10e8T^{2} \)
19 \( 1 + 2.83e4T + 8.93e8T^{2} \)
23 \( 1 - 2.08e3T + 3.40e9T^{2} \)
29 \( 1 + 1.11e5T + 1.72e10T^{2} \)
31 \( 1 - 7.97e3T + 2.75e10T^{2} \)
37 \( 1 + 1.85e5T + 9.49e10T^{2} \)
41 \( 1 + 4.21e5T + 1.94e11T^{2} \)
47 \( 1 - 1.97e5T + 5.06e11T^{2} \)
53 \( 1 + 2.01e6T + 1.17e12T^{2} \)
59 \( 1 + 8.56e5T + 2.48e12T^{2} \)
61 \( 1 - 2.61e6T + 3.14e12T^{2} \)
67 \( 1 - 2.93e6T + 6.06e12T^{2} \)
71 \( 1 + 3.20e6T + 9.09e12T^{2} \)
73 \( 1 - 4.19e6T + 1.10e13T^{2} \)
79 \( 1 - 7.83e6T + 1.92e13T^{2} \)
83 \( 1 - 5.57e6T + 2.71e13T^{2} \)
89 \( 1 + 8.89e5T + 4.42e13T^{2} \)
97 \( 1 - 1.05e7T + 8.07e13T^{2} \)
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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

−13.69333099285308359747812196242, −13.05670049335857762175995482633, −10.88900890139847141691682063440, −9.697315453160481459933010966911, −8.912541036976934438443053670321, −8.014206492790037420499803272529, −6.56287608937006667497457130747, −3.66110454699516717811952264450, −2.08999453631188021821557746347, 0, 2.08999453631188021821557746347, 3.66110454699516717811952264450, 6.56287608937006667497457130747, 8.014206492790037420499803272529, 8.912541036976934438443053670321, 9.697315453160481459933010966911, 10.88900890139847141691682063440, 13.05670049335857762175995482633, 13.69333099285308359747812196242

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