Properties

Label 2-43-1.1-c7-0-11
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  − 5.52·2-s − 57.7·3-s − 97.4·4-s + 304.·5-s + 318.·6-s + 1.42e3·7-s + 1.24e3·8-s + 1.14e3·9-s − 1.68e3·10-s − 7.14e3·11-s + 5.62e3·12-s − 1.89e3·13-s − 7.85e3·14-s − 1.75e4·15-s + 5.58e3·16-s + 1.14e3·17-s − 6.31e3·18-s − 1.61e4·19-s − 2.96e4·20-s − 8.20e4·21-s + 3.94e4·22-s + 9.16e4·23-s − 7.19e4·24-s + 1.46e4·25-s + 1.04e4·26-s + 6.02e4·27-s − 1.38e5·28-s + ⋯
L(s)  = 1  − 0.488·2-s − 1.23·3-s − 0.761·4-s + 1.08·5-s + 0.602·6-s + 1.56·7-s + 0.860·8-s + 0.522·9-s − 0.532·10-s − 1.61·11-s + 0.939·12-s − 0.239·13-s − 0.765·14-s − 1.34·15-s + 0.340·16-s + 0.0563·17-s − 0.255·18-s − 0.541·19-s − 0.829·20-s − 1.93·21-s + 0.790·22-s + 1.56·23-s − 1.06·24-s + 0.187·25-s + 0.117·26-s + 0.589·27-s − 1.19·28-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 \( 1 + 5.52T + 128T^{2} \)
3 \( 1 + 57.7T + 2.18e3T^{2} \)
5 \( 1 - 304.T + 7.81e4T^{2} \)
7 \( 1 - 1.42e3T + 8.23e5T^{2} \)
11 \( 1 + 7.14e3T + 1.94e7T^{2} \)
13 \( 1 + 1.89e3T + 6.27e7T^{2} \)
17 \( 1 - 1.14e3T + 4.10e8T^{2} \)
19 \( 1 + 1.61e4T + 8.93e8T^{2} \)
23 \( 1 - 9.16e4T + 3.40e9T^{2} \)
29 \( 1 + 2.29e5T + 1.72e10T^{2} \)
31 \( 1 + 2.12e5T + 2.75e10T^{2} \)
37 \( 1 + 4.82e5T + 9.49e10T^{2} \)
41 \( 1 + 3.84e5T + 1.94e11T^{2} \)
47 \( 1 - 3.80e5T + 5.06e11T^{2} \)
53 \( 1 + 3.61e5T + 1.17e12T^{2} \)
59 \( 1 + 1.95e6T + 2.48e12T^{2} \)
61 \( 1 + 4.79e5T + 3.14e12T^{2} \)
67 \( 1 + 2.46e6T + 6.06e12T^{2} \)
71 \( 1 + 1.56e6T + 9.09e12T^{2} \)
73 \( 1 - 2.98e6T + 1.10e13T^{2} \)
79 \( 1 - 2.64e6T + 1.92e13T^{2} \)
83 \( 1 - 5.54e6T + 2.71e13T^{2} \)
89 \( 1 + 9.15e6T + 4.42e13T^{2} \)
97 \( 1 - 4.33e6T + 8.07e13T^{2} \)
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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

−13.73142749536562453719836299172, −12.72069003825141715067163498142, −11.02261782296707208029336722075, −10.46670995267998369788312012344, −8.950632394118963464931813943362, −7.56516012059321862105248489698, −5.42933252132158105279171966710, −5.02474729284303870685524961306, −1.69707856900039546666902939838, 0, 1.69707856900039546666902939838, 5.02474729284303870685524961306, 5.42933252132158105279171966710, 7.56516012059321862105248489698, 8.950632394118963464931813943362, 10.46670995267998369788312012344, 11.02261782296707208029336722075, 12.72069003825141715067163498142, 13.73142749536562453719836299172

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