Properties

Degree 2
Conductor 43
Sign $0.819 - 0.572i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.61·2-s + (0.190 + 0.330i)3-s + 4.85·4-s + (1.61 + 2.80i)5-s + (−0.5 − 0.866i)6-s + (−0.118 + 0.204i)7-s − 7.47·8-s + (1.42 − 2.47i)9-s + (−4.23 − 7.33i)10-s − 1.38·11-s + (0.927 + 1.60i)12-s + (−1.80 + 3.13i)13-s + (0.309 − 0.535i)14-s + (−0.618 + 1.07i)15-s + 9.85·16-s + (2.54 − 4.40i)17-s + ⋯
L(s)  = 1  − 1.85·2-s + (0.110 + 0.190i)3-s + 2.42·4-s + (0.723 + 1.25i)5-s + (−0.204 − 0.353i)6-s + (−0.0446 + 0.0772i)7-s − 2.64·8-s + (0.475 − 0.823i)9-s + (−1.33 − 2.32i)10-s − 0.416·11-s + (0.267 + 0.463i)12-s + (−0.501 + 0.869i)13-s + (0.0825 − 0.143i)14-s + (−0.159 + 0.276i)15-s + 2.46·16-s + (0.617 − 1.06i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.819 - 0.572i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.819 - 0.572i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $0.819 - 0.572i$
motivic weight  =  \(1\)
character  :  $\chi_{43} (36, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 43,\ (\ :1/2),\ 0.819 - 0.572i)\)
\(L(1)\)  \(\approx\)  \(0.411383 + 0.129474i\)
\(L(\frac12)\)  \(\approx\)  \(0.411383 + 0.129474i\)
\(L(\frac{3}{2})\)   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 \( 1 + (6.5 + 0.866i)T \)
good2 \( 1 + 2.61T + 2T^{2} \)
3 \( 1 + (-0.190 - 0.330i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (-1.61 - 2.80i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (0.118 - 0.204i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + 1.38T + 11T^{2} \)
13 \( 1 + (1.80 - 3.13i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-2.54 + 4.40i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.61 + 2.80i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.30 + 5.73i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.5 - 2.59i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.927 + 1.60i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 0.527T + 41T^{2} \)
47 \( 1 - 7.85T + 47T^{2} \)
53 \( 1 + (-1.80 - 3.13i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + 6.09T + 59T^{2} \)
61 \( 1 + (-1.92 + 3.33i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.42 + 2.47i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (6.89 - 11.9i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-2.42 + 4.20i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-1.80 + 3.13i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-6.51 - 11.2i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (2.42 + 4.20i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 4.76T + 97T^{2} \)
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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

−16.40032093601856866343623148462, −15.29163884128403339793361269931, −14.23221678613015989881230786909, −12.06634139006083615744554081716, −10.75863954890462280061334070438, −9.922625935869659530103646991449, −9.065498323329843765523844033472, −7.30586229910336100940751206951, −6.46904971033821410527799815476, −2.55953159545145440458288446491, 1.74402716981569586677513667000, 5.66989695649487317765268894238, 7.65308196408449499798381789852, 8.423380481787101835574062026676, 9.790324727253329212579557074059, 10.45337744844734479329383752623, 12.21772441519814141937225684343, 13.31529566081983945513038476380, 15.32251920069512242597632947333, 16.48034449307735345195264267986

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