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  − 3.71·2-s − 24.7·3-s − 114.·4-s + 385.·5-s + 91.9·6-s − 1.54e3·7-s + 900.·8-s − 1.57e3·9-s − 1.43e3·10-s − 1.66e3·11-s + 2.82e3·12-s + 1.27e4·13-s + 5.74e3·14-s − 9.53e3·15-s + 1.12e4·16-s + 2.38e4·17-s + 5.85e3·18-s + 3.05e4·19-s − 4.40e4·20-s + 3.82e4·21-s + 6.17e3·22-s − 6.37e4·23-s − 2.22e4·24-s + 7.03e4·25-s − 4.72e4·26-s + 9.30e4·27-s + 1.76e5·28-s + ⋯
L(s)  = 1  − 0.328·2-s − 0.529·3-s − 0.892·4-s + 1.37·5-s + 0.173·6-s − 1.70·7-s + 0.621·8-s − 0.719·9-s − 0.452·10-s − 0.376·11-s + 0.472·12-s + 1.60·13-s + 0.559·14-s − 0.729·15-s + 0.687·16-s + 1.17·17-s + 0.236·18-s + 1.02·19-s − 1.22·20-s + 0.901·21-s + 0.123·22-s − 1.09·23-s − 0.328·24-s + 0.900·25-s − 0.527·26-s + 0.910·27-s + 1.52·28-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  =  \((2,\ 43,\ (\ :7/2),\ 1)\)
\(L(4)\)  \(\approx\)  \(1.00240\)
\(L(\frac12)\)  \(\approx\)  \(1.00240\)
\(L(\frac{9}{2})\)   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 \( 1 + 3.71T + 128T^{2} \)
3 \( 1 + 24.7T + 2.18e3T^{2} \)
5 \( 1 - 385.T + 7.81e4T^{2} \)
7 \( 1 + 1.54e3T + 8.23e5T^{2} \)
11 \( 1 + 1.66e3T + 1.94e7T^{2} \)
13 \( 1 - 1.27e4T + 6.27e7T^{2} \)
17 \( 1 - 2.38e4T + 4.10e8T^{2} \)
19 \( 1 - 3.05e4T + 8.93e8T^{2} \)
23 \( 1 + 6.37e4T + 3.40e9T^{2} \)
29 \( 1 - 1.90e5T + 1.72e10T^{2} \)
31 \( 1 - 1.43e5T + 2.75e10T^{2} \)
37 \( 1 + 5.38e4T + 9.49e10T^{2} \)
41 \( 1 + 3.55e5T + 1.94e11T^{2} \)
47 \( 1 + 3.94e4T + 5.06e11T^{2} \)
53 \( 1 - 1.01e6T + 1.17e12T^{2} \)
59 \( 1 - 1.24e6T + 2.48e12T^{2} \)
61 \( 1 + 7.51e5T + 3.14e12T^{2} \)
67 \( 1 + 3.22e6T + 6.06e12T^{2} \)
71 \( 1 - 1.10e6T + 9.09e12T^{2} \)
73 \( 1 - 4.53e6T + 1.10e13T^{2} \)
79 \( 1 - 3.63e6T + 1.92e13T^{2} \)
83 \( 1 + 1.49e5T + 2.71e13T^{2} \)
89 \( 1 + 8.46e6T + 4.42e13T^{2} \)
97 \( 1 - 5.99e6T + 8.07e13T^{2} \)
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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

−13.84632877057426408243208177945, −13.57820776238712133350412394739, −12.25883083815280489930045941790, −10.30615329754780594206236211774, −9.744377437670068081877593558939, −8.515800199076012361123677721784, −6.27581306900272409262278027635, −5.54155410768436745706398461625, −3.25902929607434875731701060843, −0.831624383573782639258420854093, 0.831624383573782639258420854093, 3.25902929607434875731701060843, 5.54155410768436745706398461625, 6.27581306900272409262278027635, 8.515800199076012361123677721784, 9.744377437670068081877593558939, 10.30615329754780594206236211774, 12.25883083815280489930045941790, 13.57820776238712133350412394739, 13.84632877057426408243208177945

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