Properties

Label 2-43-1.1-c5-0-17
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  + 8.95·2-s − 25.1·3-s + 48.2·4-s − 61.2·5-s − 224.·6-s − 166.·7-s + 145.·8-s + 387.·9-s − 549.·10-s + 607.·11-s − 1.21e3·12-s − 1.03e3·13-s − 1.48e3·14-s + 1.53e3·15-s − 238.·16-s + 1.43e3·17-s + 3.46e3·18-s − 1.33e3·19-s − 2.95e3·20-s + 4.16e3·21-s + 5.44e3·22-s − 437.·23-s − 3.65e3·24-s + 631.·25-s − 9.31e3·26-s − 3.62e3·27-s − 8.01e3·28-s + ⋯
L(s)  = 1  + 1.58·2-s − 1.61·3-s + 1.50·4-s − 1.09·5-s − 2.55·6-s − 1.28·7-s + 0.805·8-s + 1.59·9-s − 1.73·10-s + 1.51·11-s − 2.42·12-s − 1.70·13-s − 2.02·14-s + 1.76·15-s − 0.233·16-s + 1.20·17-s + 2.52·18-s − 0.846·19-s − 1.65·20-s + 2.06·21-s + 2.39·22-s − 0.172·23-s − 1.29·24-s + 0.202·25-s − 2.70·26-s − 0.956·27-s − 1.93·28-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 \( 1 - 8.95T + 32T^{2} \)
3 \( 1 + 25.1T + 243T^{2} \)
5 \( 1 + 61.2T + 3.12e3T^{2} \)
7 \( 1 + 166.T + 1.68e4T^{2} \)
11 \( 1 - 607.T + 1.61e5T^{2} \)
13 \( 1 + 1.03e3T + 3.71e5T^{2} \)
17 \( 1 - 1.43e3T + 1.41e6T^{2} \)
19 \( 1 + 1.33e3T + 2.47e6T^{2} \)
23 \( 1 + 437.T + 6.43e6T^{2} \)
29 \( 1 + 87.2T + 2.05e7T^{2} \)
31 \( 1 - 2.65e3T + 2.86e7T^{2} \)
37 \( 1 + 4.67e3T + 6.93e7T^{2} \)
41 \( 1 + 9.01e3T + 1.15e8T^{2} \)
47 \( 1 + 8.62e3T + 2.29e8T^{2} \)
53 \( 1 + 2.83e4T + 4.18e8T^{2} \)
59 \( 1 - 4.80e4T + 7.14e8T^{2} \)
61 \( 1 + 3.97e4T + 8.44e8T^{2} \)
67 \( 1 - 1.92e4T + 1.35e9T^{2} \)
71 \( 1 - 1.70e4T + 1.80e9T^{2} \)
73 \( 1 - 2.22e4T + 2.07e9T^{2} \)
79 \( 1 - 3.69e4T + 3.07e9T^{2} \)
83 \( 1 + 1.17e5T + 3.93e9T^{2} \)
89 \( 1 - 4.15e4T + 5.58e9T^{2} \)
97 \( 1 - 3.25e4T + 8.58e9T^{2} \)
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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

−14.41282445733040318445797365360, −12.64555570128022482923004842779, −12.16104233849528165636181436436, −11.53210726899759418637940743177, −9.915471721630831233830866232692, −7.03253982674949730631962237087, −6.20499874402488362728708371144, −4.80581840398783684321356251254, −3.61057028552080163641041369726, 0, 3.61057028552080163641041369726, 4.80581840398783684321356251254, 6.20499874402488362728708371144, 7.03253982674949730631962237087, 9.915471721630831233830866232692, 11.53210726899759418637940743177, 12.16104233849528165636181436436, 12.64555570128022482923004842779, 14.41282445733040318445797365360

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