Properties

Label 2-43-1.1-c5-0-7
Degree $2$
Conductor $43$
Sign $-1$
Analytic cond. $6.89650$
Root an. cond. $2.62611$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 7.06·2-s − 9.19·3-s + 17.8·4-s + 73.4·5-s + 64.9·6-s − 4.24·7-s + 99.7·8-s − 158.·9-s − 518.·10-s − 99.8·11-s − 164.·12-s − 441.·13-s + 29.9·14-s − 675.·15-s − 1.27e3·16-s − 1.03e3·17-s + 1.11e3·18-s + 402.·19-s + 1.31e3·20-s + 39.0·21-s + 705.·22-s − 3.45e3·23-s − 916.·24-s + 2.26e3·25-s + 3.11e3·26-s + 3.69e3·27-s − 75.9·28-s + ⋯
L(s)  = 1  − 1.24·2-s − 0.589·3-s + 0.558·4-s + 1.31·5-s + 0.736·6-s − 0.0327·7-s + 0.551·8-s − 0.652·9-s − 1.64·10-s − 0.248·11-s − 0.329·12-s − 0.724·13-s + 0.0409·14-s − 0.774·15-s − 1.24·16-s − 0.866·17-s + 0.814·18-s + 0.255·19-s + 0.733·20-s + 0.0193·21-s + 0.310·22-s − 1.36·23-s − 0.324·24-s + 0.725·25-s + 0.903·26-s + 0.974·27-s − 0.0183·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(43\)
Sign: $-1$
Analytic conductor: \(6.89650\)
Root analytic conductor: \(2.62611\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{43} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 43,\ (\ :5/2),\ -1)\)

Particular Values

\(L(3)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad43 \( 1 + 1.84e3T \)
good2 \( 1 + 7.06T + 32T^{2} \)
3 \( 1 + 9.19T + 243T^{2} \)
5 \( 1 - 73.4T + 3.12e3T^{2} \)
7 \( 1 + 4.24T + 1.68e4T^{2} \)
11 \( 1 + 99.8T + 1.61e5T^{2} \)
13 \( 1 + 441.T + 3.71e5T^{2} \)
17 \( 1 + 1.03e3T + 1.41e6T^{2} \)
19 \( 1 - 402.T + 2.47e6T^{2} \)
23 \( 1 + 3.45e3T + 6.43e6T^{2} \)
29 \( 1 + 4.26e3T + 2.05e7T^{2} \)
31 \( 1 - 3.77e3T + 2.86e7T^{2} \)
37 \( 1 + 7.57e3T + 6.93e7T^{2} \)
41 \( 1 + 1.41e4T + 1.15e8T^{2} \)
47 \( 1 + 1.34e4T + 2.29e8T^{2} \)
53 \( 1 + 1.17e4T + 4.18e8T^{2} \)
59 \( 1 - 2.39e4T + 7.14e8T^{2} \)
61 \( 1 - 4.46e4T + 8.44e8T^{2} \)
67 \( 1 - 3.92e4T + 1.35e9T^{2} \)
71 \( 1 - 6.59e3T + 1.80e9T^{2} \)
73 \( 1 + 3.63e4T + 2.07e9T^{2} \)
79 \( 1 - 6.18e4T + 3.07e9T^{2} \)
83 \( 1 + 6.46e4T + 3.93e9T^{2} \)
89 \( 1 - 2.95e4T + 5.58e9T^{2} \)
97 \( 1 - 1.93e4T + 8.58e9T^{2} \)
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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

−14.27608622106241870733812483333, −13.24369739077029664671124396138, −11.58874644881320426791901787203, −10.33217822132859680512839783167, −9.562295853632029064061444367199, −8.306715928216521818530588110818, −6.61345152876094980733448492082, −5.20896084958955493414645985591, −2.02549191619377465302371873859, 0, 2.02549191619377465302371873859, 5.20896084958955493414645985591, 6.61345152876094980733448492082, 8.306715928216521818530588110818, 9.562295853632029064061444367199, 10.33217822132859680512839783167, 11.58874644881320426791901787203, 13.24369739077029664671124396138, 14.27608622106241870733812483333

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