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Properties

Label 2-43-43.7-c2-0-2
Degree $2$
Conductor $43$
Sign $0.953 + 0.302i$
Analytic cond. $1.17166$
Root an. cond. $1.08243$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.604i·2-s + (−1.35 + 0.782i)3-s + 3.63·4-s + (4.60 − 2.65i)5-s + (0.472 + 0.818i)6-s + (−0.191 − 0.110i)7-s − 4.61i·8-s + (−3.27 + 5.67i)9-s + (−1.60 − 2.78i)10-s − 12.5·11-s + (−4.92 + 2.84i)12-s + (−3.33 + 5.77i)13-s + (−0.0667 + 0.115i)14-s + (−4.15 + 7.20i)15-s + 11.7·16-s + (4.41 − 7.65i)17-s + ⋯
L(s)  = 1  − 0.302i·2-s + (−0.451 + 0.260i)3-s + 0.908·4-s + (0.920 − 0.531i)5-s + (0.0787 + 0.136i)6-s + (−0.0273 − 0.0157i)7-s − 0.576i·8-s + (−0.364 + 0.630i)9-s + (−0.160 − 0.278i)10-s − 1.14·11-s + (−0.410 + 0.236i)12-s + (−0.256 + 0.444i)13-s + (−0.00476 + 0.00825i)14-s + (−0.277 + 0.480i)15-s + 0.734·16-s + (0.259 − 0.450i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.953 + 0.302i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.953 + 0.302i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(43\)
Sign: $0.953 + 0.302i$
Analytic conductor: \(1.17166\)
Root analytic conductor: \(1.08243\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{43} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 43,\ (\ :1),\ 0.953 + 0.302i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.16865 - 0.180826i\)
\(L(\frac12)\) \(\approx\) \(1.16865 - 0.180826i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad43 \( 1 + (16.4 - 39.7i)T \)
good2 \( 1 + 0.604iT - 4T^{2} \)
3 \( 1 + (1.35 - 0.782i)T + (4.5 - 7.79i)T^{2} \)
5 \( 1 + (-4.60 + 2.65i)T + (12.5 - 21.6i)T^{2} \)
7 \( 1 + (0.191 + 0.110i)T + (24.5 + 42.4i)T^{2} \)
11 \( 1 + 12.5T + 121T^{2} \)
13 \( 1 + (3.33 - 5.77i)T + (-84.5 - 146. i)T^{2} \)
17 \( 1 + (-4.41 + 7.65i)T + (-144.5 - 250. i)T^{2} \)
19 \( 1 + (20.3 - 11.7i)T + (180.5 - 312. i)T^{2} \)
23 \( 1 + (14.5 + 25.1i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + (-26.8 - 15.5i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (-17.3 - 29.9i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + (-35.4 + 20.4i)T + (684.5 - 1.18e3i)T^{2} \)
41 \( 1 + 18.0T + 1.68e3T^{2} \)
47 \( 1 - 35.2T + 2.20e3T^{2} \)
53 \( 1 + (35.9 + 62.2i)T + (-1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 - 22.2T + 3.48e3T^{2} \)
61 \( 1 + (-65.8 - 37.9i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (34.2 + 59.2i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + (-12.9 - 7.45i)T + (2.52e3 + 4.36e3i)T^{2} \)
73 \( 1 + (-40.2 - 23.2i)T + (2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-40.1 + 69.5i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (-22.6 - 39.3i)T + (-3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 + (-93.4 + 53.9i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 - 99.3T + 9.40e3T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−16.11512233114184917288192025622, −14.50585414143147521736905650242, −13.16438680717050518479206358389, −12.08750621698547552406971366408, −10.73859479455104151153556481761, −10.01395985595169590180031252127, −8.171591761903025679251882092802, −6.36723625943508012379282669194, −5.06709250393216061567207889353, −2.32328628597949626376632539785, 2.56817783622168817653848119549, 5.69203858728574384985268007630, 6.50639607606919975737852262359, 7.973658091191342192289172694358, 9.971198360510089185167296125306, 10.97306812553160399867769082742, 12.20270639491946959162955733611, 13.49142455402450708184709344845, 14.86807057925633773567111151079, 15.66495291429380702836404962883

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