Properties

Degree 2
Conductor 43
Sign $1$
Motivic weight 5
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 10.7·2-s + 18.3·3-s + 82.6·4-s − 72.1·5-s + 196.·6-s − 96.4·7-s + 542.·8-s + 93.5·9-s − 772.·10-s − 684.·11-s + 1.51e3·12-s + 344.·13-s − 1.03e3·14-s − 1.32e3·15-s + 3.15e3·16-s + 1.31e3·17-s + 1.00e3·18-s + 739.·19-s − 5.96e3·20-s − 1.76e3·21-s − 7.32e3·22-s + 3.16e3·23-s + 9.94e3·24-s + 2.08e3·25-s + 3.68e3·26-s − 2.74e3·27-s − 7.97e3·28-s + ⋯
L(s)  = 1  + 1.89·2-s + 1.17·3-s + 2.58·4-s − 1.29·5-s + 2.22·6-s − 0.744·7-s + 2.99·8-s + 0.384·9-s − 2.44·10-s − 1.70·11-s + 3.03·12-s + 0.565·13-s − 1.40·14-s − 1.51·15-s + 3.08·16-s + 1.10·17-s + 0.728·18-s + 0.469·19-s − 3.33·20-s − 0.875·21-s − 3.22·22-s + 1.24·23-s + 3.52·24-s + 0.667·25-s + 1.07·26-s − 0.723·27-s − 1.92·28-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

\( d \)  =  \(2\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(5\)
character  :  $\chi_{43} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 43,\ (\ :5/2),\ 1)\)
\(L(3)\)  \(\approx\)  \(5.00289\)
\(L(\frac12)\)  \(\approx\)  \(5.00289\)
\(L(\frac{7}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \neq 43$,\(F_p(T)\) is a polynomial of degree 2. If $p = 43$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad43 \( 1 - 1.84e3T \)
good2 \( 1 - 10.7T + 32T^{2} \)
3 \( 1 - 18.3T + 243T^{2} \)
5 \( 1 + 72.1T + 3.12e3T^{2} \)
7 \( 1 + 96.4T + 1.68e4T^{2} \)
11 \( 1 + 684.T + 1.61e5T^{2} \)
13 \( 1 - 344.T + 3.71e5T^{2} \)
17 \( 1 - 1.31e3T + 1.41e6T^{2} \)
19 \( 1 - 739.T + 2.47e6T^{2} \)
23 \( 1 - 3.16e3T + 6.43e6T^{2} \)
29 \( 1 - 7.07e3T + 2.05e7T^{2} \)
31 \( 1 + 3.79e3T + 2.86e7T^{2} \)
37 \( 1 + 1.26e4T + 6.93e7T^{2} \)
41 \( 1 + 1.08e4T + 1.15e8T^{2} \)
47 \( 1 - 3.87e3T + 2.29e8T^{2} \)
53 \( 1 - 6.47e3T + 4.18e8T^{2} \)
59 \( 1 - 3.47e4T + 7.14e8T^{2} \)
61 \( 1 - 2.64e4T + 8.44e8T^{2} \)
67 \( 1 + 5.80e4T + 1.35e9T^{2} \)
71 \( 1 + 2.34e4T + 1.80e9T^{2} \)
73 \( 1 - 4.51e4T + 2.07e9T^{2} \)
79 \( 1 - 1.78e4T + 3.07e9T^{2} \)
83 \( 1 - 3.97e4T + 3.93e9T^{2} \)
89 \( 1 + 3.08e4T + 5.58e9T^{2} \)
97 \( 1 - 2.25e4T + 8.58e9T^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−14.85369084352915453764885303399, −13.72133159195809562023772543790, −12.92649505079387964038378517102, −11.89013864721557129175314374684, −10.52127923753517581748192909480, −8.188840301281531922291314899889, −7.15586871550439614723143666380, −5.24867699807912780449493787938, −3.55919740872722686252924413259, −2.91965384678829062864018954373, 2.91965384678829062864018954373, 3.55919740872722686252924413259, 5.24867699807912780449493787938, 7.15586871550439614723143666380, 8.188840301281531922291314899889, 10.52127923753517581748192909480, 11.89013864721557129175314374684, 12.92649505079387964038378517102, 13.72133159195809562023772543790, 14.85369084352915453764885303399

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