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  + 43.0·2-s − 179.·3-s + 1.34e3·4-s − 1.33e3·5-s − 7.74e3·6-s − 743.·7-s + 3.58e4·8-s + 1.25e4·9-s − 5.73e4·10-s − 6.08e4·11-s − 2.41e5·12-s − 1.04e5·13-s − 3.20e4·14-s + 2.39e5·15-s + 8.58e5·16-s − 3.44e5·17-s + 5.42e5·18-s − 6.26e5·19-s − 1.79e6·20-s + 1.33e5·21-s − 2.62e6·22-s + 1.13e6·23-s − 6.44e6·24-s − 1.81e5·25-s − 4.48e6·26-s + 1.27e6·27-s − 1.00e6·28-s + ⋯
L(s)  = 1  + 1.90·2-s − 1.28·3-s + 2.62·4-s − 0.952·5-s − 2.43·6-s − 0.117·7-s + 3.09·8-s + 0.639·9-s − 1.81·10-s − 1.25·11-s − 3.36·12-s − 1.01·13-s − 0.222·14-s + 1.21·15-s + 3.27·16-s − 1.00·17-s + 1.21·18-s − 1.10·19-s − 2.50·20-s + 0.149·21-s − 2.38·22-s + 0.842·23-s − 3.96·24-s − 0.0929·25-s − 1.92·26-s + 0.461·27-s − 0.307·28-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 \( 1 - 43.0T + 512T^{2} \)
3 \( 1 + 179.T + 1.96e4T^{2} \)
5 \( 1 + 1.33e3T + 1.95e6T^{2} \)
7 \( 1 + 743.T + 4.03e7T^{2} \)
11 \( 1 + 6.08e4T + 2.35e9T^{2} \)
13 \( 1 + 1.04e5T + 1.06e10T^{2} \)
17 \( 1 + 3.44e5T + 1.18e11T^{2} \)
19 \( 1 + 6.26e5T + 3.22e11T^{2} \)
23 \( 1 - 1.13e6T + 1.80e12T^{2} \)
29 \( 1 - 3.51e6T + 1.45e13T^{2} \)
31 \( 1 + 6.97e6T + 2.64e13T^{2} \)
37 \( 1 - 7.60e6T + 1.29e14T^{2} \)
41 \( 1 - 3.04e7T + 3.27e14T^{2} \)
47 \( 1 - 3.57e7T + 1.11e15T^{2} \)
53 \( 1 + 9.41e7T + 3.29e15T^{2} \)
59 \( 1 + 1.25e8T + 8.66e15T^{2} \)
61 \( 1 - 1.62e8T + 1.16e16T^{2} \)
67 \( 1 - 1.76e8T + 2.72e16T^{2} \)
71 \( 1 + 7.75e7T + 4.58e16T^{2} \)
73 \( 1 + 4.14e8T + 5.88e16T^{2} \)
79 \( 1 - 3.87e8T + 1.19e17T^{2} \)
83 \( 1 - 1.61e7T + 1.86e17T^{2} \)
89 \( 1 + 1.53e8T + 3.50e17T^{2} \)
97 \( 1 - 1.02e9T + 7.60e17T^{2} \)
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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

−12.97226371502031092875924535707, −12.43652799223122361322918841931, −11.29371961958079190168654725414, −10.73162694466302996795991773090, −7.57222150168529014589665737364, −6.41415143732155559022284911039, −5.17207223835358917663761685416, −4.33873463014208192143088009410, −2.60399065431517269471434369553, 0, 2.60399065431517269471434369553, 4.33873463014208192143088009410, 5.17207223835358917663761685416, 6.41415143732155559022284911039, 7.57222150168529014589665737364, 10.73162694466302996795991773090, 11.29371961958079190168654725414, 12.43652799223122361322918841931, 12.97226371502031092875924535707

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