Properties

Label 2-43-1.1-c5-0-9
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  − 8.09·2-s + 11.1·3-s + 33.5·4-s − 63.3·5-s − 90.5·6-s + 223.·7-s − 12.3·8-s − 118.·9-s + 513.·10-s − 631.·11-s + 374.·12-s + 28.5·13-s − 1.80e3·14-s − 708.·15-s − 972.·16-s − 1.74e3·17-s + 955.·18-s − 2.02e3·19-s − 2.12e3·20-s + 2.49e3·21-s + 5.11e3·22-s + 2.98e3·23-s − 138.·24-s + 891.·25-s − 231.·26-s − 4.03e3·27-s + 7.49e3·28-s + ⋯
L(s)  = 1  − 1.43·2-s + 0.717·3-s + 1.04·4-s − 1.13·5-s − 1.02·6-s + 1.72·7-s − 0.0684·8-s − 0.485·9-s + 1.62·10-s − 1.57·11-s + 0.751·12-s + 0.0468·13-s − 2.46·14-s − 0.813·15-s − 0.949·16-s − 1.46·17-s + 0.694·18-s − 1.28·19-s − 1.18·20-s + 1.23·21-s + 2.25·22-s + 1.17·23-s − 0.0490·24-s + 0.285·25-s − 0.0671·26-s − 1.06·27-s + 1.80·28-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 \( 1 + 8.09T + 32T^{2} \)
3 \( 1 - 11.1T + 243T^{2} \)
5 \( 1 + 63.3T + 3.12e3T^{2} \)
7 \( 1 - 223.T + 1.68e4T^{2} \)
11 \( 1 + 631.T + 1.61e5T^{2} \)
13 \( 1 - 28.5T + 3.71e5T^{2} \)
17 \( 1 + 1.74e3T + 1.41e6T^{2} \)
19 \( 1 + 2.02e3T + 2.47e6T^{2} \)
23 \( 1 - 2.98e3T + 6.43e6T^{2} \)
29 \( 1 - 766.T + 2.05e7T^{2} \)
31 \( 1 + 8.35e3T + 2.86e7T^{2} \)
37 \( 1 - 1.48e4T + 6.93e7T^{2} \)
41 \( 1 + 5.34e3T + 1.15e8T^{2} \)
47 \( 1 + 6.28e3T + 2.29e8T^{2} \)
53 \( 1 + 915.T + 4.18e8T^{2} \)
59 \( 1 + 1.46e4T + 7.14e8T^{2} \)
61 \( 1 + 2.13e4T + 8.44e8T^{2} \)
67 \( 1 + 1.28e4T + 1.35e9T^{2} \)
71 \( 1 - 5.64e4T + 1.80e9T^{2} \)
73 \( 1 + 2.55e4T + 2.07e9T^{2} \)
79 \( 1 - 5.79e3T + 3.07e9T^{2} \)
83 \( 1 + 7.85e3T + 3.93e9T^{2} \)
89 \( 1 + 7.56e3T + 5.58e9T^{2} \)
97 \( 1 - 1.11e5T + 8.58e9T^{2} \)
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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

−14.81266172438715342228282959546, −13.23696644054165401898759475617, −11.20985925006358875922831813832, −10.90652350799077099010902092986, −8.867343471671081749398651881828, −8.188782739072086039119712820882, −7.53104743647850313451639363339, −4.64256204116743641268691058528, −2.21761255570618475923519179815, 0, 2.21761255570618475923519179815, 4.64256204116743641268691058528, 7.53104743647850313451639363339, 8.188782739072086039119712820882, 8.867343471671081749398651881828, 10.90652350799077099010902092986, 11.20985925006358875922831813832, 13.23696644054165401898759475617, 14.81266172438715342228282959546

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