Properties

Degree 2
Conductor 43
Sign $1$
Motivic weight 7
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 22.1·2-s + 2.16·3-s + 364.·4-s + 122.·5-s + 47.9·6-s − 90.8·7-s + 5.23e3·8-s − 2.18e3·9-s + 2.71e3·10-s + 3.41e3·11-s + 787.·12-s − 1.12e4·13-s − 2.01e3·14-s + 265.·15-s + 6.95e4·16-s + 9.72e3·17-s − 4.84e4·18-s + 2.53e4·19-s + 4.46e4·20-s − 196.·21-s + 7.56e4·22-s − 5.27e4·23-s + 1.13e4·24-s − 6.31e4·25-s − 2.50e5·26-s − 9.45e3·27-s − 3.30e4·28-s + ⋯
L(s)  = 1  + 1.96·2-s + 0.0462·3-s + 2.84·4-s + 0.438·5-s + 0.0907·6-s − 0.100·7-s + 3.61·8-s − 0.997·9-s + 0.859·10-s + 0.772·11-s + 0.131·12-s − 1.42·13-s − 0.196·14-s + 0.0202·15-s + 4.24·16-s + 0.479·17-s − 1.95·18-s + 0.847·19-s + 1.24·20-s − 0.00463·21-s + 1.51·22-s − 0.904·23-s + 0.167·24-s − 0.807·25-s − 2.79·26-s − 0.0924·27-s − 0.284·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(7\)
character  :  $\chi_{43} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 43,\ (\ :7/2),\ 1)\)
\(L(4)\)  \(\approx\)  \(5.99700\)
\(L(\frac12)\)  \(\approx\)  \(5.99700\)
\(L(\frac{9}{2})\)   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 \( 1 + 7.95e4T \)
good2 \( 1 - 22.1T + 128T^{2} \)
3 \( 1 - 2.16T + 2.18e3T^{2} \)
5 \( 1 - 122.T + 7.81e4T^{2} \)
7 \( 1 + 90.8T + 8.23e5T^{2} \)
11 \( 1 - 3.41e3T + 1.94e7T^{2} \)
13 \( 1 + 1.12e4T + 6.27e7T^{2} \)
17 \( 1 - 9.72e3T + 4.10e8T^{2} \)
19 \( 1 - 2.53e4T + 8.93e8T^{2} \)
23 \( 1 + 5.27e4T + 3.40e9T^{2} \)
29 \( 1 + 2.16e5T + 1.72e10T^{2} \)
31 \( 1 - 1.42e5T + 2.75e10T^{2} \)
37 \( 1 + 7.42e4T + 9.49e10T^{2} \)
41 \( 1 + 2.73e5T + 1.94e11T^{2} \)
47 \( 1 + 1.12e5T + 5.06e11T^{2} \)
53 \( 1 - 1.41e6T + 1.17e12T^{2} \)
59 \( 1 + 1.44e6T + 2.48e12T^{2} \)
61 \( 1 - 3.19e6T + 3.14e12T^{2} \)
67 \( 1 + 3.52e6T + 6.06e12T^{2} \)
71 \( 1 + 3.27e5T + 9.09e12T^{2} \)
73 \( 1 - 2.80e6T + 1.10e13T^{2} \)
79 \( 1 - 3.35e6T + 1.92e13T^{2} \)
83 \( 1 - 7.10e6T + 2.71e13T^{2} \)
89 \( 1 + 1.32e6T + 4.42e13T^{2} \)
97 \( 1 - 1.29e7T + 8.07e13T^{2} \)
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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

−14.30769620983766743227110710375, −13.52102793473375979449285696154, −12.15161572206610553612411153889, −11.55242865958215995634596897426, −9.908769981668025279408377928051, −7.53977719545711111866302064619, −6.13447964086615433943317570100, −5.13410915226749823730529571096, −3.52553292236940281300784223832, −2.13827213695954792577595187478, 2.13827213695954792577595187478, 3.52553292236940281300784223832, 5.13410915226749823730529571096, 6.13447964086615433943317570100, 7.53977719545711111866302064619, 9.908769981668025279408377928051, 11.55242865958215995634596897426, 12.15161572206610553612411153889, 13.52102793473375979449285696154, 14.30769620983766743227110710375

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