Properties

Degree 2
Conductor 43
Sign $1$
Motivic weight 3
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.84·2-s + 9.49·3-s − 4.58·4-s + 2.98·5-s + 17.5·6-s − 26.0·7-s − 23.2·8-s + 63.1·9-s + 5.51·10-s − 36.8·11-s − 43.5·12-s + 89.5·13-s − 48.1·14-s + 28.3·15-s − 6.28·16-s − 28.8·17-s + 116.·18-s − 58.8·19-s − 13.6·20-s − 247.·21-s − 68.0·22-s + 2.63·23-s − 220.·24-s − 116.·25-s + 165.·26-s + 343.·27-s + 119.·28-s + ⋯
L(s)  = 1  + 0.653·2-s + 1.82·3-s − 0.573·4-s + 0.266·5-s + 1.19·6-s − 1.40·7-s − 1.02·8-s + 2.34·9-s + 0.174·10-s − 1.01·11-s − 1.04·12-s + 1.91·13-s − 0.919·14-s + 0.487·15-s − 0.0981·16-s − 0.410·17-s + 1.52·18-s − 0.710·19-s − 0.152·20-s − 2.57·21-s − 0.659·22-s + 0.0238·23-s − 1.87·24-s − 0.928·25-s + 1.24·26-s + 2.44·27-s + 0.806·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(3\)
character  :  $\chi_{43} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 43,\ (\ :3/2),\ 1)\)
\(L(2)\)  \(\approx\)  \(2.32228\)
\(L(\frac12)\)  \(\approx\)  \(2.32228\)
\(L(\frac{5}{2})\)   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 \( 1 + 43T \)
good2 \( 1 - 1.84T + 8T^{2} \)
3 \( 1 - 9.49T + 27T^{2} \)
5 \( 1 - 2.98T + 125T^{2} \)
7 \( 1 + 26.0T + 343T^{2} \)
11 \( 1 + 36.8T + 1.33e3T^{2} \)
13 \( 1 - 89.5T + 2.19e3T^{2} \)
17 \( 1 + 28.8T + 4.91e3T^{2} \)
19 \( 1 + 58.8T + 6.85e3T^{2} \)
23 \( 1 - 2.63T + 1.21e4T^{2} \)
29 \( 1 - 173.T + 2.43e4T^{2} \)
31 \( 1 - 57.9T + 2.97e4T^{2} \)
37 \( 1 - 52.0T + 5.06e4T^{2} \)
41 \( 1 - 142.T + 6.89e4T^{2} \)
47 \( 1 + 106.T + 1.03e5T^{2} \)
53 \( 1 - 244.T + 1.48e5T^{2} \)
59 \( 1 + 127.T + 2.05e5T^{2} \)
61 \( 1 + 443.T + 2.26e5T^{2} \)
67 \( 1 + 117.T + 3.00e5T^{2} \)
71 \( 1 - 816.T + 3.57e5T^{2} \)
73 \( 1 + 620.T + 3.89e5T^{2} \)
79 \( 1 - 377.T + 4.93e5T^{2} \)
83 \( 1 + 1.45e3T + 5.71e5T^{2} \)
89 \( 1 - 627.T + 7.04e5T^{2} \)
97 \( 1 + 817.T + 9.12e5T^{2} \)
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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

−15.35410033160672927268256430029, −13.93195792384677763996598150110, −13.35425629029052116932541128721, −12.80975565625599376603045010698, −10.20451753931685064210670780806, −9.128084058298134759691398236464, −8.253588136707017446862521317260, −6.28400983251440728734163494312, −3.99448284606684095514476385754, −2.90431880886791947813790758202, 2.90431880886791947813790758202, 3.99448284606684095514476385754, 6.28400983251440728734163494312, 8.253588136707017446862521317260, 9.128084058298134759691398236464, 10.20451753931685064210670780806, 12.80975565625599376603045010698, 13.35425629029052116932541128721, 13.93195792384677763996598150110, 15.35410033160672927268256430029

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