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  − 0.381·2-s + (1.30 − 2.26i)3-s − 1.85·4-s + (−0.618 + 1.07i)5-s + (−0.5 + 0.866i)6-s + (2.11 + 3.66i)7-s + 1.47·8-s + (−1.92 − 3.33i)9-s + (0.236 − 0.408i)10-s − 3.61·11-s + (−2.42 + 4.20i)12-s + (−0.690 − 1.19i)13-s + (−0.809 − 1.40i)14-s + (1.61 + 2.80i)15-s + 3.14·16-s + (−3.04 − 5.27i)17-s + ⋯
L(s)  = 1  − 0.270·2-s + (0.755 − 1.30i)3-s − 0.927·4-s + (−0.276 + 0.478i)5-s + (−0.204 + 0.353i)6-s + (0.800 + 1.38i)7-s + 0.520·8-s + (−0.642 − 1.11i)9-s + (0.0746 − 0.129i)10-s − 1.09·11-s + (−0.700 + 1.21i)12-s + (−0.191 − 0.331i)13-s + (−0.216 − 0.374i)14-s + (0.417 + 0.723i)15-s + 0.786·16-s + (−0.738 − 1.27i)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  :  $\chi_{43} (6, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 43,\ (\ :1/2),\ 0.819 + 0.572i)\)
\(L(1)\)  \(\approx\)  \(0.694667 - 0.218632i\)
\(L(\frac12)\)  \(\approx\)  \(0.694667 - 0.218632i\)
\(L(\frac{3}{2})\)   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 \( 1 + 0.381T + 2T^{2} \)
3 \( 1 + (-1.30 + 2.26i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (0.618 - 1.07i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (-2.11 - 3.66i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + 3.61T + 11T^{2} \)
13 \( 1 + (0.690 + 1.19i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (3.04 + 5.27i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.618 + 1.07i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.19 - 3.79i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.5 + 2.59i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.42 + 4.20i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 9.47T + 41T^{2} \)
47 \( 1 - 1.14T + 47T^{2} \)
53 \( 1 + (-0.690 + 1.19i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 5.09T + 59T^{2} \)
61 \( 1 + (1.42 + 2.47i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.92 + 3.33i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-5.39 - 9.35i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (0.927 + 1.60i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.690 - 1.19i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (8.01 - 13.8i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-0.927 + 1.60i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 9.23T + 97T^{2} \)
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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

−15.59660990333386405580324516386, −14.55380390153089915960820691798, −13.54300432608014054711072834601, −12.64585042453488935203674863028, −11.34106048485691358533374741013, −9.349533236810762881900709650631, −8.260519378923592803666677811258, −7.44950380377139430208752767735, −5.31054274513364251844579521823, −2.54138790800309779570144036565, 4.06187580240804674038035488591, 4.74613885977079331036065932448, 7.926082290983334950019780376527, 8.685150939867275425868089843298, 10.10430762237744544771690344315, 10.72338412949536074580214522873, 12.95156210073968578539620623176, 14.04945194580653810387808516450, 14.85893551766919761044864105123, 16.20715498279783986557702774064

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