Properties

Label 2-43-1.1-c7-0-8
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  − 16.1·2-s − 48.3·3-s + 133.·4-s − 272.·5-s + 780.·6-s + 1.10e3·7-s − 81.7·8-s + 148.·9-s + 4.40e3·10-s + 685.·11-s − 6.42e3·12-s + 1.32e4·13-s − 1.77e4·14-s + 1.31e4·15-s − 1.57e4·16-s − 1.15e3·17-s − 2.39e3·18-s − 1.29e4·19-s − 3.63e4·20-s − 5.32e4·21-s − 1.10e4·22-s − 1.06e5·23-s + 3.94e3·24-s − 3.63e3·25-s − 2.13e5·26-s + 9.85e4·27-s + 1.46e5·28-s + ⋯
L(s)  = 1  − 1.42·2-s − 1.03·3-s + 1.03·4-s − 0.976·5-s + 1.47·6-s + 1.21·7-s − 0.0564·8-s + 0.0677·9-s + 1.39·10-s + 0.155·11-s − 1.07·12-s + 1.66·13-s − 1.73·14-s + 1.00·15-s − 0.958·16-s − 0.0570·17-s − 0.0966·18-s − 0.433·19-s − 1.01·20-s − 1.25·21-s − 0.221·22-s − 1.83·23-s + 0.0582·24-s − 0.0464·25-s − 2.38·26-s + 0.963·27-s + 1.26·28-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 \( 1 + 16.1T + 128T^{2} \)
3 \( 1 + 48.3T + 2.18e3T^{2} \)
5 \( 1 + 272.T + 7.81e4T^{2} \)
7 \( 1 - 1.10e3T + 8.23e5T^{2} \)
11 \( 1 - 685.T + 1.94e7T^{2} \)
13 \( 1 - 1.32e4T + 6.27e7T^{2} \)
17 \( 1 + 1.15e3T + 4.10e8T^{2} \)
19 \( 1 + 1.29e4T + 8.93e8T^{2} \)
23 \( 1 + 1.06e5T + 3.40e9T^{2} \)
29 \( 1 - 7.14e4T + 1.72e10T^{2} \)
31 \( 1 - 2.29e5T + 2.75e10T^{2} \)
37 \( 1 + 2.79e5T + 9.49e10T^{2} \)
41 \( 1 - 3.45e5T + 1.94e11T^{2} \)
47 \( 1 + 7.35e5T + 5.06e11T^{2} \)
53 \( 1 + 7.14e5T + 1.17e12T^{2} \)
59 \( 1 + 9.15e5T + 2.48e12T^{2} \)
61 \( 1 + 575.T + 3.14e12T^{2} \)
67 \( 1 + 3.55e6T + 6.06e12T^{2} \)
71 \( 1 + 1.66e6T + 9.09e12T^{2} \)
73 \( 1 + 1.03e6T + 1.10e13T^{2} \)
79 \( 1 - 1.58e6T + 1.92e13T^{2} \)
83 \( 1 + 8.26e6T + 2.71e13T^{2} \)
89 \( 1 - 7.66e6T + 4.42e13T^{2} \)
97 \( 1 + 1.26e7T + 8.07e13T^{2} \)
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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

−13.96609857819406390462297178478, −11.89300415138124070475039985190, −11.27903326424657361550394725260, −10.43167810473048490506021469891, −8.548904058447261788140538795977, −7.937189985566166330894510636149, −6.25195478361896149052804814142, −4.38407571352645288159798361556, −1.35380385121915730978728954136, 0, 1.35380385121915730978728954136, 4.38407571352645288159798361556, 6.25195478361896149052804814142, 7.937189985566166330894510636149, 8.548904058447261788140538795977, 10.43167810473048490506021469891, 11.27903326424657361550394725260, 11.89300415138124070475039985190, 13.96609857819406390462297178478

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