Properties

Degree 2
Conductor 43
Sign $-0.147 + 0.989i$
Motivic weight 6
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.61i·2-s + (−35.7 − 20.6i)3-s + 57.1·4-s + (121. + 70.2i)5-s + (−53.9 + 93.4i)6-s + (140. − 81.1i)7-s − 316. i·8-s + (486. + 841. i)9-s + (183. − 318. i)10-s + 633.·11-s + (−2.04e3 − 1.17e3i)12-s + (−2.02e3 − 3.51e3i)13-s + (−212. − 367. i)14-s + (−2.89e3 − 5.02e3i)15-s + 2.82e3·16-s + (−1.52e3 − 2.64e3i)17-s + ⋯
L(s)  = 1  − 0.326i·2-s + (−1.32 − 0.763i)3-s + 0.893·4-s + (0.973 + 0.562i)5-s + (−0.249 + 0.432i)6-s + (0.410 − 0.236i)7-s − 0.618i·8-s + (0.666 + 1.15i)9-s + (0.183 − 0.318i)10-s + 0.475·11-s + (−1.18 − 0.682i)12-s + (−0.923 − 1.59i)13-s + (−0.0773 − 0.134i)14-s + (−0.858 − 1.48i)15-s + 0.690·16-s + (−0.310 − 0.538i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.147 + 0.989i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.147 + 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $-0.147 + 0.989i$
motivic weight  =  \(6\)
character  :  $\chi_{43} (37, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 43,\ (\ :3),\ -0.147 + 0.989i)\)
\(L(\frac{7}{2})\)  \(\approx\)  \(1.03541 - 1.20082i\)
\(L(\frac12)\)  \(\approx\)  \(1.03541 - 1.20082i\)
\(L(4)\)   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 \( 1 + (-7.93e4 - 4.88e3i)T \)
good2 \( 1 + 2.61iT - 64T^{2} \)
3 \( 1 + (35.7 + 20.6i)T + (364.5 + 631. i)T^{2} \)
5 \( 1 + (-121. - 70.2i)T + (7.81e3 + 1.35e4i)T^{2} \)
7 \( 1 + (-140. + 81.1i)T + (5.88e4 - 1.01e5i)T^{2} \)
11 \( 1 - 633.T + 1.77e6T^{2} \)
13 \( 1 + (2.02e3 + 3.51e3i)T + (-2.41e6 + 4.18e6i)T^{2} \)
17 \( 1 + (1.52e3 + 2.64e3i)T + (-1.20e7 + 2.09e7i)T^{2} \)
19 \( 1 + (-1.71e3 - 990. i)T + (2.35e7 + 4.07e7i)T^{2} \)
23 \( 1 + (-2.76e3 + 4.79e3i)T + (-7.40e7 - 1.28e8i)T^{2} \)
29 \( 1 + (-1.91e4 + 1.10e4i)T + (2.97e8 - 5.15e8i)T^{2} \)
31 \( 1 + (-445. + 770. i)T + (-4.43e8 - 7.68e8i)T^{2} \)
37 \( 1 + (3.29e4 + 1.90e4i)T + (1.28e9 + 2.22e9i)T^{2} \)
41 \( 1 + 5.31e4T + 4.75e9T^{2} \)
47 \( 1 - 1.44e5T + 1.07e10T^{2} \)
53 \( 1 + (-1.26e4 + 2.19e4i)T + (-1.10e10 - 1.91e10i)T^{2} \)
59 \( 1 + 1.68e5T + 4.21e10T^{2} \)
61 \( 1 + (-2.78e5 + 1.61e5i)T + (2.57e10 - 4.46e10i)T^{2} \)
67 \( 1 + (2.20e5 - 3.81e5i)T + (-4.52e10 - 7.83e10i)T^{2} \)
71 \( 1 + (-4.35e5 + 2.51e5i)T + (6.40e10 - 1.10e11i)T^{2} \)
73 \( 1 + (5.79e5 - 3.34e5i)T + (7.56e10 - 1.31e11i)T^{2} \)
79 \( 1 + (-4.84e5 - 8.39e5i)T + (-1.21e11 + 2.10e11i)T^{2} \)
83 \( 1 + (2.27e5 - 3.94e5i)T + (-1.63e11 - 2.83e11i)T^{2} \)
89 \( 1 + (8.33e5 + 4.81e5i)T + (2.48e11 + 4.30e11i)T^{2} \)
97 \( 1 - 5.54e5T + 8.32e11T^{2} \)
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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

−14.24586255406298896673784658787, −12.81050750710104392535813398931, −11.93673078474215034294047123456, −10.85635670597632369320659620680, −10.09096916914186521833089845755, −7.47776174836816854000405800764, −6.48648164208754223643078886148, −5.43284355164062857070613084016, −2.48283808964133321384689361171, −0.910556586961999959624838230924, 1.74809105755675952415053103058, 4.71761652423436962383888809419, 5.78171007932280858384388457362, 6.86940794940780926474383340191, 9.085790895169604885613910838968, 10.30463814139867667533275710531, 11.48789016527187483164443256175, 12.19240324304177753909630663043, 14.06879936396435194695717216833, 15.30789311466600947607327865563

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