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  + 5.38·2-s + 25.0·3-s − 2.96·4-s − 0.456·5-s + 134.·6-s + 166.·7-s − 188.·8-s + 384.·9-s − 2.46·10-s − 65.6·11-s − 74.3·12-s − 689.·13-s + 897.·14-s − 11.4·15-s − 920.·16-s + 737.·17-s + 2.07e3·18-s − 609.·19-s + 1.35·20-s + 4.17e3·21-s − 353.·22-s + 1.31e3·23-s − 4.71e3·24-s − 3.12e3·25-s − 3.71e3·26-s + 3.53e3·27-s − 494.·28-s + ⋯
L(s)  = 1  + 0.952·2-s + 1.60·3-s − 0.0927·4-s − 0.00816·5-s + 1.53·6-s + 1.28·7-s − 1.04·8-s + 1.58·9-s − 0.00778·10-s − 0.163·11-s − 0.148·12-s − 1.13·13-s + 1.22·14-s − 0.0131·15-s − 0.898·16-s + 0.618·17-s + 1.50·18-s − 0.387·19-s + 0.000757·20-s + 2.06·21-s − 0.155·22-s + 0.517·23-s − 1.67·24-s − 0.999·25-s − 1.07·26-s + 0.934·27-s − 0.119·28-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  =  \((2,\ 43,\ (\ :5/2),\ 1)\)
\(L(3)\)  \(\approx\)  \(3.73317\)
\(L(\frac12)\)  \(\approx\)  \(3.73317\)
\(L(\frac{7}{2})\)   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 \( 1 - 5.38T + 32T^{2} \)
3 \( 1 - 25.0T + 243T^{2} \)
5 \( 1 + 0.456T + 3.12e3T^{2} \)
7 \( 1 - 166.T + 1.68e4T^{2} \)
11 \( 1 + 65.6T + 1.61e5T^{2} \)
13 \( 1 + 689.T + 3.71e5T^{2} \)
17 \( 1 - 737.T + 1.41e6T^{2} \)
19 \( 1 + 609.T + 2.47e6T^{2} \)
23 \( 1 - 1.31e3T + 6.43e6T^{2} \)
29 \( 1 + 8.96e3T + 2.05e7T^{2} \)
31 \( 1 - 5.85e3T + 2.86e7T^{2} \)
37 \( 1 + 55.9T + 6.93e7T^{2} \)
41 \( 1 + 1.04e4T + 1.15e8T^{2} \)
47 \( 1 - 1.94e3T + 2.29e8T^{2} \)
53 \( 1 - 3.01e4T + 4.18e8T^{2} \)
59 \( 1 - 5.21e4T + 7.14e8T^{2} \)
61 \( 1 + 1.40e4T + 8.44e8T^{2} \)
67 \( 1 - 5.14e4T + 1.35e9T^{2} \)
71 \( 1 - 1.12e4T + 1.80e9T^{2} \)
73 \( 1 + 6.11e4T + 2.07e9T^{2} \)
79 \( 1 - 9.60e4T + 3.07e9T^{2} \)
83 \( 1 - 3.03e4T + 3.93e9T^{2} \)
89 \( 1 + 6.50e4T + 5.58e9T^{2} \)
97 \( 1 - 1.20e4T + 8.58e9T^{2} \)
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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

−14.76759364650930155151865137894, −13.96494143573098622042316360875, −13.04685000624374343511186970905, −11.74582059281931756877415550339, −9.792859320056461187143944657344, −8.586377372274699065151568744793, −7.55084003112198253140383718022, −5.15950162687004251845581096357, −3.84167185072649783030299320982, −2.28343938478795840589969228638, 2.28343938478795840589969228638, 3.84167185072649783030299320982, 5.15950162687004251845581096357, 7.55084003112198253140383718022, 8.586377372274699065151568744793, 9.792859320056461187143944657344, 11.74582059281931756877415550339, 13.04685000624374343511186970905, 13.96494143573098622042316360875, 14.76759364650930155151865137894

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