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  − 19.1·2-s − 36.5·3-s + 239.·4-s − 174.·5-s + 701.·6-s − 1.12e3·7-s − 2.13e3·8-s − 848.·9-s + 3.34e3·10-s − 8.17e3·11-s − 8.75e3·12-s − 1.35e4·13-s + 2.15e4·14-s + 6.39e3·15-s + 1.02e4·16-s + 3.75e4·17-s + 1.62e4·18-s − 2.66e4·19-s − 4.18e4·20-s + 4.12e4·21-s + 1.56e5·22-s − 2.00e4·23-s + 7.80e4·24-s − 4.75e4·25-s + 2.58e5·26-s + 1.11e5·27-s − 2.69e5·28-s + ⋯
L(s)  = 1  − 1.69·2-s − 0.782·3-s + 1.86·4-s − 0.625·5-s + 1.32·6-s − 1.24·7-s − 1.47·8-s − 0.387·9-s + 1.05·10-s − 1.85·11-s − 1.46·12-s − 1.70·13-s + 2.10·14-s + 0.489·15-s + 0.626·16-s + 1.85·17-s + 0.657·18-s − 0.890·19-s − 1.16·20-s + 0.971·21-s + 3.13·22-s − 0.343·23-s + 1.15·24-s − 0.609·25-s + 2.88·26-s + 1.08·27-s − 2.32·28-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  =  \((2,\ 43,\ (\ :7/2),\ 1)\)
\(L(4)\)  \(\approx\)  \(0.00863577\)
\(L(\frac12)\)  \(\approx\)  \(0.00863577\)
\(L(\frac{9}{2})\)   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 \( 1 + 19.1T + 128T^{2} \)
3 \( 1 + 36.5T + 2.18e3T^{2} \)
5 \( 1 + 174.T + 7.81e4T^{2} \)
7 \( 1 + 1.12e3T + 8.23e5T^{2} \)
11 \( 1 + 8.17e3T + 1.94e7T^{2} \)
13 \( 1 + 1.35e4T + 6.27e7T^{2} \)
17 \( 1 - 3.75e4T + 4.10e8T^{2} \)
19 \( 1 + 2.66e4T + 8.93e8T^{2} \)
23 \( 1 + 2.00e4T + 3.40e9T^{2} \)
29 \( 1 + 2.75e4T + 1.72e10T^{2} \)
31 \( 1 + 1.84e5T + 2.75e10T^{2} \)
37 \( 1 - 3.01e5T + 9.49e10T^{2} \)
41 \( 1 + 5.07e5T + 1.94e11T^{2} \)
47 \( 1 + 1.37e5T + 5.06e11T^{2} \)
53 \( 1 - 5.77e5T + 1.17e12T^{2} \)
59 \( 1 - 1.70e6T + 2.48e12T^{2} \)
61 \( 1 - 1.24e6T + 3.14e12T^{2} \)
67 \( 1 + 1.24e6T + 6.06e12T^{2} \)
71 \( 1 + 3.10e6T + 9.09e12T^{2} \)
73 \( 1 + 3.60e6T + 1.10e13T^{2} \)
79 \( 1 + 3.14e6T + 1.92e13T^{2} \)
83 \( 1 - 3.47e5T + 2.71e13T^{2} \)
89 \( 1 + 4.60e6T + 4.42e13T^{2} \)
97 \( 1 + 1.55e7T + 8.07e13T^{2} \)
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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

−15.02229244497209198637899267857, −12.74511134491466713415993734738, −11.75158072779074583124558126913, −10.39183323479207734380024709705, −9.819869896339698228289667518825, −8.113976615784479540481691326651, −7.20970961911341860228868405158, −5.57301701407736829814352041662, −2.72418982127281489097211794682, −0.085078150271300746480761381589, 0.085078150271300746480761381589, 2.72418982127281489097211794682, 5.57301701407736829814352041662, 7.20970961911341860228868405158, 8.113976615784479540481691326651, 9.819869896339698228289667518825, 10.39183323479207734380024709705, 11.75158072779074583124558126913, 12.74511134491466713415993734738, 15.02229244497209198637899267857

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