Properties

Degree 2
Conductor 43
Sign $-0.330 - 0.943i$
Motivic weight 3
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.62 − 2.03i)2-s + (−6.93 − 1.04i)3-s + (0.276 − 1.21i)4-s + (6.69 + 4.56i)5-s + (9.11 + 15.7i)6-s + (−16.5 + 28.6i)7-s + (−21.6 + 10.4i)8-s + (21.1 + 6.52i)9-s + (−1.57 − 21.0i)10-s + (0.540 + 2.36i)11-s + (−3.18 + 8.11i)12-s + (4.87 − 64.9i)13-s + (85.0 − 12.8i)14-s + (−41.6 − 38.6i)15-s + (47.3 + 22.7i)16-s + (−78.9 + 53.8i)17-s + ⋯
L(s)  = 1  + (−0.572 − 0.718i)2-s + (−1.33 − 0.201i)3-s + (0.0345 − 0.151i)4-s + (0.599 + 0.408i)5-s + (0.619 + 1.07i)6-s + (−0.893 + 1.54i)7-s + (−0.956 + 0.460i)8-s + (0.783 + 0.241i)9-s + (−0.0497 − 0.664i)10-s + (0.0148 + 0.0648i)11-s + (−0.0766 + 0.195i)12-s + (0.103 − 1.38i)13-s + (1.62 − 0.244i)14-s + (−0.717 − 0.665i)15-s + (0.739 + 0.355i)16-s + (−1.12 + 0.768i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.330 - 0.943i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.330 - 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $-0.330 - 0.943i$
motivic weight  =  \(3\)
character  :  $\chi_{43} (9, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 43,\ (\ :3/2),\ -0.330 - 0.943i)\)
\(L(2)\)  \(\approx\)  \(0.0703233 + 0.0991030i\)
\(L(\frac12)\)  \(\approx\)  \(0.0703233 + 0.0991030i\)
\(L(\frac{5}{2})\)   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 \( 1 + (-259. + 111. i)T \)
good2 \( 1 + (1.62 + 2.03i)T + (-1.78 + 7.79i)T^{2} \)
3 \( 1 + (6.93 + 1.04i)T + (25.8 + 7.95i)T^{2} \)
5 \( 1 + (-6.69 - 4.56i)T + (45.6 + 116. i)T^{2} \)
7 \( 1 + (16.5 - 28.6i)T + (-171.5 - 297. i)T^{2} \)
11 \( 1 + (-0.540 - 2.36i)T + (-1.19e3 + 577. i)T^{2} \)
13 \( 1 + (-4.87 + 64.9i)T + (-2.17e3 - 327. i)T^{2} \)
17 \( 1 + (78.9 - 53.8i)T + (1.79e3 - 4.57e3i)T^{2} \)
19 \( 1 + (98.5 - 30.3i)T + (5.66e3 - 3.86e3i)T^{2} \)
23 \( 1 + (71.4 - 66.2i)T + (909. - 1.21e4i)T^{2} \)
29 \( 1 + (-5.06 + 0.764i)T + (2.33e4 - 7.18e3i)T^{2} \)
31 \( 1 + (-41.3 + 105. i)T + (-2.18e4 - 2.02e4i)T^{2} \)
37 \( 1 + (145. + 252. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (-136. - 170. i)T + (-1.53e4 + 6.71e4i)T^{2} \)
47 \( 1 + (26.9 - 117. i)T + (-9.35e4 - 4.50e4i)T^{2} \)
53 \( 1 + (-25.3 - 338. i)T + (-1.47e5 + 2.21e4i)T^{2} \)
59 \( 1 + (17.3 + 8.36i)T + (1.28e5 + 1.60e5i)T^{2} \)
61 \( 1 + (-184. - 470. i)T + (-1.66e5 + 1.54e5i)T^{2} \)
67 \( 1 + (371. - 114. i)T + (2.48e5 - 1.69e5i)T^{2} \)
71 \( 1 + (638. + 592. i)T + (2.67e4 + 3.56e5i)T^{2} \)
73 \( 1 + (34.5 - 460. i)T + (-3.84e5 - 5.79e4i)T^{2} \)
79 \( 1 + (-7.37 + 12.7i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (712. + 107. i)T + (5.46e5 + 1.68e5i)T^{2} \)
89 \( 1 + (-481. - 72.6i)T + (6.73e5 + 2.07e5i)T^{2} \)
97 \( 1 + (-89.0 - 390. i)T + (-8.22e5 + 3.95e5i)T^{2} \)
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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

−15.84330812421954785688594828972, −14.96009962529338721768550644989, −12.87058456288191838174574664095, −12.11561178115531101832510012273, −10.90154758641991585755969989687, −10.11066371249099885304971744287, −8.793073301098294581691332159721, −6.03710686254329825504459777407, −5.92249076769582353165512436198, −2.38237878435077612572062412341, 0.12792910419528377439181452888, 4.37617489244562917026733277617, 6.37295680926903238486974589541, 6.92962372078708824529366744169, 8.966178789975260251132246977134, 10.16254138146655422344733312549, 11.39083987910296208326624416190, 12.75183347154123259889231466369, 13.84967065016714307612006019850, 15.89774355537251328438325845474

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