Properties

Label 2-43-1.1-c5-0-4
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  − 11.1·2-s − 28.1·3-s + 92.7·4-s − 10.1·5-s + 314.·6-s + 135.·7-s − 678.·8-s + 550.·9-s + 113.·10-s − 74.1·11-s − 2.61e3·12-s − 252.·13-s − 1.51e3·14-s + 285.·15-s + 4.60e3·16-s + 233.·17-s − 6.15e3·18-s + 550.·19-s − 941.·20-s − 3.82e3·21-s + 827.·22-s + 1.95e3·23-s + 1.91e4·24-s − 3.02e3·25-s + 2.81e3·26-s − 8.67e3·27-s + 1.25e4·28-s + ⋯
L(s)  = 1  − 1.97·2-s − 1.80·3-s + 2.89·4-s − 0.181·5-s + 3.56·6-s + 1.04·7-s − 3.74·8-s + 2.26·9-s + 0.358·10-s − 0.184·11-s − 5.23·12-s − 0.414·13-s − 2.06·14-s + 0.328·15-s + 4.49·16-s + 0.195·17-s − 4.47·18-s + 0.349·19-s − 0.526·20-s − 1.89·21-s + 0.364·22-s + 0.769·23-s + 6.77·24-s − 0.967·25-s + 0.817·26-s − 2.29·27-s + 3.03·28-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 \( 1 + 11.1T + 32T^{2} \)
3 \( 1 + 28.1T + 243T^{2} \)
5 \( 1 + 10.1T + 3.12e3T^{2} \)
7 \( 1 - 135.T + 1.68e4T^{2} \)
11 \( 1 + 74.1T + 1.61e5T^{2} \)
13 \( 1 + 252.T + 3.71e5T^{2} \)
17 \( 1 - 233.T + 1.41e6T^{2} \)
19 \( 1 - 550.T + 2.47e6T^{2} \)
23 \( 1 - 1.95e3T + 6.43e6T^{2} \)
29 \( 1 + 4.67e3T + 2.05e7T^{2} \)
31 \( 1 - 3.33e3T + 2.86e7T^{2} \)
37 \( 1 + 1.34e4T + 6.93e7T^{2} \)
41 \( 1 - 8.20e3T + 1.15e8T^{2} \)
47 \( 1 + 2.20e4T + 2.29e8T^{2} \)
53 \( 1 - 6.34e3T + 4.18e8T^{2} \)
59 \( 1 + 1.49e4T + 7.14e8T^{2} \)
61 \( 1 + 2.38e4T + 8.44e8T^{2} \)
67 \( 1 - 3.86e4T + 1.35e9T^{2} \)
71 \( 1 - 4.91e3T + 1.80e9T^{2} \)
73 \( 1 - 5.26e3T + 2.07e9T^{2} \)
79 \( 1 + 2.68e4T + 3.07e9T^{2} \)
83 \( 1 - 4.84e4T + 3.93e9T^{2} \)
89 \( 1 + 1.17e5T + 5.58e9T^{2} \)
97 \( 1 + 6.23e4T + 8.58e9T^{2} \)
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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

−15.16024332511933762586842397141, −12.25633217373522056255784082412, −11.43295136109433833789952891292, −10.77790597690493720787243743792, −9.663352230951310471161401524871, −7.941011043460991876130080519491, −6.88495390871003755932359170181, −5.44441463934444830842461284647, −1.46870562390291089684251473883, 0, 1.46870562390291089684251473883, 5.44441463934444830842461284647, 6.88495390871003755932359170181, 7.941011043460991876130080519491, 9.663352230951310471161401524871, 10.77790597690493720787243743792, 11.43295136109433833789952891292, 12.25633217373522056255784082412, 15.16024332511933762586842397141

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