Properties

Label 2-43-1.1-c5-0-16
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  + 3.65·2-s + 7.84·3-s − 18.6·4-s − 107.·5-s + 28.6·6-s − 25.5·7-s − 185.·8-s − 181.·9-s − 391.·10-s + 512.·11-s − 146.·12-s + 862.·13-s − 93.3·14-s − 840.·15-s − 81.0·16-s − 1.52e3·17-s − 663.·18-s − 1.54e3·19-s + 1.99e3·20-s − 200.·21-s + 1.87e3·22-s − 3.12e3·23-s − 1.45e3·24-s + 8.34e3·25-s + 3.15e3·26-s − 3.32e3·27-s + 475.·28-s + ⋯
L(s)  = 1  + 0.646·2-s + 0.503·3-s − 0.582·4-s − 1.91·5-s + 0.325·6-s − 0.196·7-s − 1.02·8-s − 0.746·9-s − 1.23·10-s + 1.27·11-s − 0.292·12-s + 1.41·13-s − 0.127·14-s − 0.963·15-s − 0.0791·16-s − 1.27·17-s − 0.482·18-s − 0.980·19-s + 1.11·20-s − 0.0990·21-s + 0.824·22-s − 1.23·23-s − 0.514·24-s + 2.67·25-s + 0.915·26-s − 0.878·27-s + 0.114·28-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 \( 1 - 3.65T + 32T^{2} \)
3 \( 1 - 7.84T + 243T^{2} \)
5 \( 1 + 107.T + 3.12e3T^{2} \)
7 \( 1 + 25.5T + 1.68e4T^{2} \)
11 \( 1 - 512.T + 1.61e5T^{2} \)
13 \( 1 - 862.T + 3.71e5T^{2} \)
17 \( 1 + 1.52e3T + 1.41e6T^{2} \)
19 \( 1 + 1.54e3T + 2.47e6T^{2} \)
23 \( 1 + 3.12e3T + 6.43e6T^{2} \)
29 \( 1 + 947.T + 2.05e7T^{2} \)
31 \( 1 - 339.T + 2.86e7T^{2} \)
37 \( 1 + 7.44e3T + 6.93e7T^{2} \)
41 \( 1 - 5.11e3T + 1.15e8T^{2} \)
47 \( 1 + 1.71e4T + 2.29e8T^{2} \)
53 \( 1 - 1.80e4T + 4.18e8T^{2} \)
59 \( 1 - 1.70e4T + 7.14e8T^{2} \)
61 \( 1 - 1.06e4T + 8.44e8T^{2} \)
67 \( 1 + 8.79e3T + 1.35e9T^{2} \)
71 \( 1 + 7.70e4T + 1.80e9T^{2} \)
73 \( 1 + 7.96e3T + 2.07e9T^{2} \)
79 \( 1 - 6.89e4T + 3.07e9T^{2} \)
83 \( 1 - 4.08e4T + 3.93e9T^{2} \)
89 \( 1 + 8.36e4T + 5.58e9T^{2} \)
97 \( 1 - 1.59e5T + 8.58e9T^{2} \)
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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

−14.45387182419655208377532076189, −13.28843617047927260693618146862, −12.01302149409927521918018992556, −11.19809853709363450681087323551, −8.862872858298842281310293493031, −8.341452557504761906696294681406, −6.41699306793740470880002334647, −4.21569658407090260443429138100, −3.55094668721447519004578440936, 0, 3.55094668721447519004578440936, 4.21569658407090260443429138100, 6.41699306793740470880002334647, 8.341452557504761906696294681406, 8.862872858298842281310293493031, 11.19809853709363450681087323551, 12.01302149409927521918018992556, 13.28843617047927260693618146862, 14.45387182419655208377532076189

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