Properties

Degree 2
Conductor 43
Sign $-0.197 - 0.980i$
Motivic weight 10
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5.51i·2-s + 292. i·3-s + 993.·4-s − 3.20e3i·5-s − 1.61e3·6-s + 1.65e4i·7-s + 1.11e4i·8-s − 2.67e4·9-s + 1.76e4·10-s + 3.15e4·11-s + 2.90e5i·12-s + 2.40e5·13-s − 9.10e4·14-s + 9.38e5·15-s + 9.56e5·16-s + 1.02e6·17-s + ⋯
L(s)  = 1  + 0.172i·2-s + 1.20i·3-s + 0.970·4-s − 1.02i·5-s − 0.207·6-s + 0.982i·7-s + 0.339i·8-s − 0.452·9-s + 0.176·10-s + 0.196·11-s + 1.16i·12-s + 0.646·13-s − 0.169·14-s + 1.23·15-s + 0.911·16-s + 0.723·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.197 - 0.980i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (-0.197 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $-0.197 - 0.980i$
motivic weight  =  \(10\)
character  :  $\chi_{43} (42, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 43,\ (\ :5),\ -0.197 - 0.980i)\)
\(L(\frac{11}{2})\)  \(\approx\)  \(1.64834 + 2.01307i\)
\(L(\frac12)\)  \(\approx\)  \(1.64834 + 2.01307i\)
\(L(6)\)   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 \( 1 + (-2.90e7 - 1.44e8i)T \)
good2 \( 1 - 5.51iT - 1.02e3T^{2} \)
3 \( 1 - 292. iT - 5.90e4T^{2} \)
5 \( 1 + 3.20e3iT - 9.76e6T^{2} \)
7 \( 1 - 1.65e4iT - 2.82e8T^{2} \)
11 \( 1 - 3.15e4T + 2.59e10T^{2} \)
13 \( 1 - 2.40e5T + 1.37e11T^{2} \)
17 \( 1 - 1.02e6T + 2.01e12T^{2} \)
19 \( 1 - 1.67e6iT - 6.13e12T^{2} \)
23 \( 1 + 6.08e6T + 4.14e13T^{2} \)
29 \( 1 - 1.23e7iT - 4.20e14T^{2} \)
31 \( 1 + 5.08e6T + 8.19e14T^{2} \)
37 \( 1 - 2.30e7iT - 4.80e15T^{2} \)
41 \( 1 + 7.74e7T + 1.34e16T^{2} \)
47 \( 1 + 2.22e8T + 5.25e16T^{2} \)
53 \( 1 + 1.87e8T + 1.74e17T^{2} \)
59 \( 1 + 5.73e8T + 5.11e17T^{2} \)
61 \( 1 - 4.87e7iT - 7.13e17T^{2} \)
67 \( 1 - 6.11e8T + 1.82e18T^{2} \)
71 \( 1 + 2.16e9iT - 3.25e18T^{2} \)
73 \( 1 - 5.90e8iT - 4.29e18T^{2} \)
79 \( 1 - 3.70e9T + 9.46e18T^{2} \)
83 \( 1 - 4.86e9T + 1.55e19T^{2} \)
89 \( 1 + 6.61e9iT - 3.11e19T^{2} \)
97 \( 1 - 3.91e9T + 7.37e19T^{2} \)
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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

−14.50991076253474179346929529503, −12.61928970884649689127718323652, −11.69596372447519500179604452501, −10.39743034805960630842340611771, −9.225309340828950779814408030237, −8.098121369228500701752421463293, −6.08002166719105451341338815040, −4.99866959931267792570343872633, −3.41335940717006964194139015941, −1.57157701752995067998164040924, 0.877861647262700336472932672793, 2.11924837575118649406163093076, 3.52722088933423811893283740617, 6.24922391192616224171945796676, 7.01573305834523971647291821666, 7.83771102526070773744104381345, 10.17405398034777853776165082793, 11.11310896409288890889673808593, 12.15505657397960373074973713902, 13.41387254006604936600466853223

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