Properties

Label 2-43-1.1-c7-0-20
Degree $2$
Conductor $43$
Sign $-1$
Analytic cond. $13.4325$
Root an. cond. $3.66504$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 12.3·2-s − 46.7·3-s + 23.7·4-s + 330.·5-s − 575.·6-s − 467.·7-s − 1.28e3·8-s − 4.26·9-s + 4.06e3·10-s − 359.·11-s − 1.10e3·12-s − 1.14e4·13-s − 5.75e3·14-s − 1.54e4·15-s − 1.88e4·16-s − 1.90e4·17-s − 52.5·18-s − 3.67e3·19-s + 7.83e3·20-s + 2.18e4·21-s − 4.42e3·22-s − 3.78e3·23-s + 6.00e4·24-s + 3.08e4·25-s − 1.40e5·26-s + 1.02e5·27-s − 1.10e4·28-s + ⋯
L(s)  = 1  + 1.08·2-s − 0.999·3-s + 0.185·4-s + 1.18·5-s − 1.08·6-s − 0.515·7-s − 0.886·8-s − 0.00195·9-s + 1.28·10-s − 0.0813·11-s − 0.185·12-s − 1.44·13-s − 0.560·14-s − 1.18·15-s − 1.15·16-s − 0.937·17-s − 0.00212·18-s − 0.122·19-s + 0.219·20-s + 0.514·21-s − 0.0886·22-s − 0.0648·23-s + 0.886·24-s + 0.395·25-s − 1.57·26-s + 1.00·27-s − 0.0955·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(43\)
Sign: $-1$
Analytic conductor: \(13.4325\)
Root analytic conductor: \(3.66504\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{43} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 43,\ (\ :7/2),\ -1)\)

Particular Values

\(L(4)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad43 \( 1 - 7.95e4T \)
good2 \( 1 - 12.3T + 128T^{2} \)
3 \( 1 + 46.7T + 2.18e3T^{2} \)
5 \( 1 - 330.T + 7.81e4T^{2} \)
7 \( 1 + 467.T + 8.23e5T^{2} \)
11 \( 1 + 359.T + 1.94e7T^{2} \)
13 \( 1 + 1.14e4T + 6.27e7T^{2} \)
17 \( 1 + 1.90e4T + 4.10e8T^{2} \)
19 \( 1 + 3.67e3T + 8.93e8T^{2} \)
23 \( 1 + 3.78e3T + 3.40e9T^{2} \)
29 \( 1 - 1.15e5T + 1.72e10T^{2} \)
31 \( 1 - 1.51e4T + 2.75e10T^{2} \)
37 \( 1 + 3.57e5T + 9.49e10T^{2} \)
41 \( 1 - 5.40e5T + 1.94e11T^{2} \)
47 \( 1 - 2.89e5T + 5.06e11T^{2} \)
53 \( 1 + 3.29e5T + 1.17e12T^{2} \)
59 \( 1 - 2.44e6T + 2.48e12T^{2} \)
61 \( 1 + 1.25e6T + 3.14e12T^{2} \)
67 \( 1 + 1.70e6T + 6.06e12T^{2} \)
71 \( 1 + 3.75e6T + 9.09e12T^{2} \)
73 \( 1 + 5.83e6T + 1.10e13T^{2} \)
79 \( 1 - 3.48e6T + 1.92e13T^{2} \)
83 \( 1 - 5.17e6T + 2.71e13T^{2} \)
89 \( 1 - 5.98e6T + 4.42e13T^{2} \)
97 \( 1 + 5.18e6T + 8.07e13T^{2} \)
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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

−13.73917945048380543205527869263, −12.75335571506111587990225556098, −11.85693168664428954731389403640, −10.34955893482261682711443353588, −9.164023073046718888968380791942, −6.64003393596984893501336132629, −5.70676012729189905859340193146, −4.71022872187411868986256535883, −2.59540400657507680730583755510, 0, 2.59540400657507680730583755510, 4.71022872187411868986256535883, 5.70676012729189905859340193146, 6.64003393596984893501336132629, 9.164023073046718888968380791942, 10.34955893482261682711443353588, 11.85693168664428954731389403640, 12.75335571506111587990225556098, 13.73917945048380543205527869263

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