Properties

Degree 2
Conductor 43
Sign $-0.183 + 0.983i$
Motivic weight 10
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 41.1i·2-s + 18.8i·3-s − 671.·4-s − 4.21e3i·5-s − 775.·6-s + 2.03e4i·7-s + 1.45e4i·8-s + 5.86e4·9-s + 1.73e5·10-s − 3.06e5·11-s − 1.26e4i·12-s + 1.98e5·13-s − 8.39e5·14-s + 7.93e4·15-s − 1.28e6·16-s − 2.08e6·17-s + ⋯
L(s)  = 1  + 1.28i·2-s + 0.0774i·3-s − 0.656·4-s − 1.34i·5-s − 0.0997·6-s + 1.21i·7-s + 0.442i·8-s + 0.993·9-s + 1.73·10-s − 1.90·11-s − 0.0508i·12-s + 0.535·13-s − 1.56·14-s + 0.104·15-s − 1.22·16-s − 1.47·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.183 + 0.983i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (-0.183 + 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $-0.183 + 0.983i$
motivic weight  =  \(10\)
character  :  $\chi_{43} (42, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 43,\ (\ :5),\ -0.183 + 0.983i)\)
\(L(\frac{11}{2})\)  \(\approx\)  \(0.0797310 - 0.0959706i\)
\(L(\frac12)\)  \(\approx\)  \(0.0797310 - 0.0959706i\)
\(L(6)\)   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 \( 1 + (-2.69e7 + 1.44e8i)T \)
good2 \( 1 - 41.1iT - 1.02e3T^{2} \)
3 \( 1 - 18.8iT - 5.90e4T^{2} \)
5 \( 1 + 4.21e3iT - 9.76e6T^{2} \)
7 \( 1 - 2.03e4iT - 2.82e8T^{2} \)
11 \( 1 + 3.06e5T + 2.59e10T^{2} \)
13 \( 1 - 1.98e5T + 1.37e11T^{2} \)
17 \( 1 + 2.08e6T + 2.01e12T^{2} \)
19 \( 1 + 2.97e6iT - 6.13e12T^{2} \)
23 \( 1 - 1.17e6T + 4.14e13T^{2} \)
29 \( 1 - 2.43e7iT - 4.20e14T^{2} \)
31 \( 1 + 3.70e7T + 8.19e14T^{2} \)
37 \( 1 + 7.72e7iT - 4.80e15T^{2} \)
41 \( 1 + 6.69e7T + 1.34e16T^{2} \)
47 \( 1 + 4.29e8T + 5.25e16T^{2} \)
53 \( 1 - 2.00e8T + 1.74e17T^{2} \)
59 \( 1 - 3.24e8T + 5.11e17T^{2} \)
61 \( 1 + 8.68e8iT - 7.13e17T^{2} \)
67 \( 1 + 1.36e9T + 1.82e18T^{2} \)
71 \( 1 - 2.65e9iT - 3.25e18T^{2} \)
73 \( 1 + 1.07e7iT - 4.29e18T^{2} \)
79 \( 1 - 8.01e8T + 9.46e18T^{2} \)
83 \( 1 + 5.22e9T + 1.55e19T^{2} \)
89 \( 1 - 6.11e9iT - 3.11e19T^{2} \)
97 \( 1 - 3.80e8T + 7.37e19T^{2} \)
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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

−15.19606191678428262753022063138, −13.26686091730578416490518892006, −12.76056462767299726719961971540, −11.01650122137862452168501940617, −9.079385778203615549884155459635, −8.442719312279161030974769362956, −7.06716809042456705555304558909, −5.44274566602196088250124177104, −4.83877697308018681902182011863, −2.14899054309842127612108908535, 0.03694832313173052287741712882, 1.76689741916456752335935606903, 3.05128274043596067437649687782, 4.24276116499711059433060228810, 6.65254496809663475281976327312, 7.69036956779483391022967150137, 10.07743197682113782451542182677, 10.49446950721986566223194234263, 11.29407108965051908993672024226, 13.04896591419626843782041743477

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