Properties

Degree 2
Conductor 43
Sign $-0.548 + 0.835i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3.18i·2-s + 0.724i·3-s − 6.17·4-s − 7.66i·5-s + 2.31·6-s + 10.1i·7-s + 6.93i·8-s + 8.47·9-s − 24.4·10-s − 2.43·11-s − 4.47i·12-s + 18.1·13-s + 32.3·14-s + 5.55·15-s − 2.56·16-s − 1.13·17-s + ⋯
L(s)  = 1  − 1.59i·2-s + 0.241i·3-s − 1.54·4-s − 1.53i·5-s + 0.385·6-s + 1.44i·7-s + 0.867i·8-s + 0.941·9-s − 2.44·10-s − 0.221·11-s − 0.372i·12-s + 1.39·13-s + 2.30·14-s + 0.370·15-s − 0.160·16-s − 0.0668·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.548 + 0.835i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.548 + 0.835i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $-0.548 + 0.835i$
motivic weight  =  \(2\)
character  :  $\chi_{43} (42, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 43,\ (\ :1),\ -0.548 + 0.835i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.508398 - 0.942177i\)
\(L(\frac12)\)  \(\approx\)  \(0.508398 - 0.942177i\)
\(L(2)\)   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 \( 1 + (-23.6 + 35.9i)T \)
good2 \( 1 + 3.18iT - 4T^{2} \)
3 \( 1 - 0.724iT - 9T^{2} \)
5 \( 1 + 7.66iT - 25T^{2} \)
7 \( 1 - 10.1iT - 49T^{2} \)
11 \( 1 + 2.43T + 121T^{2} \)
13 \( 1 - 18.1T + 169T^{2} \)
17 \( 1 + 1.13T + 289T^{2} \)
19 \( 1 - 12.3iT - 361T^{2} \)
23 \( 1 + 24.2T + 529T^{2} \)
29 \( 1 - 35.5iT - 841T^{2} \)
31 \( 1 + 12.9T + 961T^{2} \)
37 \( 1 + 41.5iT - 1.36e3T^{2} \)
41 \( 1 + 65.0T + 1.68e3T^{2} \)
47 \( 1 - 51.0T + 2.20e3T^{2} \)
53 \( 1 - 56.2T + 2.80e3T^{2} \)
59 \( 1 + 88.8T + 3.48e3T^{2} \)
61 \( 1 - 65.1iT - 3.72e3T^{2} \)
67 \( 1 - 45.3T + 4.48e3T^{2} \)
71 \( 1 + 63.7iT - 5.04e3T^{2} \)
73 \( 1 + 8.80iT - 5.32e3T^{2} \)
79 \( 1 - 31.8T + 6.24e3T^{2} \)
83 \( 1 + 68.4T + 6.88e3T^{2} \)
89 \( 1 - 5.90iT - 7.92e3T^{2} \)
97 \( 1 + 100.T + 9.40e3T^{2} \)
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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

−15.58477659947896799062421316842, −13.53657218177087252897099965035, −12.53479604982410125782912073732, −12.08262585553958769693851959855, −10.59167370490043841868817313334, −9.266206650096693770906066391707, −8.588204109117985501052580824096, −5.45903340111289014273680447667, −3.96291013845544232158186303894, −1.65841435123056432683290311835, 4.05029658287028016652525092089, 6.32326823478957323977116468781, 7.05165582860087801247629754626, 7.973921695708962753195757935974, 10.03517640499595481846533351122, 11.08759172015533384725513832655, 13.51832605971891400059162528119, 13.85678581426571478671542068310, 15.18901798823119060397294870095, 15.90837262134642793302888387730

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