Properties

Degree 2
Conductor 43
Sign $-1$
Motivic weight 9
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 17.9·2-s − 172.·3-s − 190.·4-s + 238.·5-s + 3.09e3·6-s − 4.52e3·7-s + 1.25e4·8-s + 1.02e4·9-s − 4.27e3·10-s + 6.67e4·11-s + 3.30e4·12-s + 4.55e4·13-s + 8.10e4·14-s − 4.12e4·15-s − 1.28e5·16-s + 4.94e5·17-s − 1.83e5·18-s − 1.08e6·19-s − 4.55e4·20-s + 7.82e5·21-s − 1.19e6·22-s + 1.35e6·23-s − 2.17e6·24-s − 1.89e6·25-s − 8.16e5·26-s + 1.63e6·27-s + 8.63e5·28-s + ⋯
L(s)  = 1  − 0.791·2-s − 1.23·3-s − 0.372·4-s + 0.170·5-s + 0.976·6-s − 0.712·7-s + 1.08·8-s + 0.520·9-s − 0.135·10-s + 1.37·11-s + 0.459·12-s + 0.442·13-s + 0.564·14-s − 0.210·15-s − 0.488·16-s + 1.43·17-s − 0.412·18-s − 1.90·19-s − 0.0636·20-s + 0.878·21-s − 1.08·22-s + 1.00·23-s − 1.34·24-s − 0.970·25-s − 0.350·26-s + 0.591·27-s + 0.265·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(9\)
character  :  $\chi_{43} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(1\)
Selberg data  =  \((2,\ 43,\ (\ :9/2),\ -1)\)
\(L(5)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{11}{2})\)   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 \( 1 + 3.41e6T \)
good2 \( 1 + 17.9T + 512T^{2} \)
3 \( 1 + 172.T + 1.96e4T^{2} \)
5 \( 1 - 238.T + 1.95e6T^{2} \)
7 \( 1 + 4.52e3T + 4.03e7T^{2} \)
11 \( 1 - 6.67e4T + 2.35e9T^{2} \)
13 \( 1 - 4.55e4T + 1.06e10T^{2} \)
17 \( 1 - 4.94e5T + 1.18e11T^{2} \)
19 \( 1 + 1.08e6T + 3.22e11T^{2} \)
23 \( 1 - 1.35e6T + 1.80e12T^{2} \)
29 \( 1 + 4.85e6T + 1.45e13T^{2} \)
31 \( 1 - 5.69e6T + 2.64e13T^{2} \)
37 \( 1 + 1.59e7T + 1.29e14T^{2} \)
41 \( 1 - 1.10e7T + 3.27e14T^{2} \)
47 \( 1 - 2.98e7T + 1.11e15T^{2} \)
53 \( 1 - 7.43e6T + 3.29e15T^{2} \)
59 \( 1 - 1.08e8T + 8.66e15T^{2} \)
61 \( 1 - 1.99e8T + 1.16e16T^{2} \)
67 \( 1 + 2.61e8T + 2.72e16T^{2} \)
71 \( 1 - 6.58e7T + 4.58e16T^{2} \)
73 \( 1 + 4.01e8T + 5.88e16T^{2} \)
79 \( 1 - 1.23e8T + 1.19e17T^{2} \)
83 \( 1 + 3.81e8T + 1.86e17T^{2} \)
89 \( 1 + 3.81e8T + 3.50e17T^{2} \)
97 \( 1 + 5.93e8T + 7.60e17T^{2} \)
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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

−13.17484947654419959451848815380, −12.02214128663562919940712552176, −10.79329624492640039263667562686, −9.770321466158423282962252786657, −8.619610888214610715268722784223, −6.82967202988138605458525052504, −5.68351981353853908603099673564, −4.01369569334513901494302797873, −1.23348914030851825508734487266, 0, 1.23348914030851825508734487266, 4.01369569334513901494302797873, 5.68351981353853908603099673564, 6.82967202988138605458525052504, 8.619610888214610715268722784223, 9.770321466158423282962252786657, 10.79329624492640039263667562686, 12.02214128663562919940712552176, 13.17484947654419959451848815380

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