Properties

Label 2-43-43.22-c2-0-3
Degree $2$
Conductor $43$
Sign $0.940 + 0.339i$
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  + (−0.468 + 0.972i)2-s + (−1.93 − 4.01i)3-s + (1.76 + 2.21i)4-s + (8.30 − 1.89i)5-s + 4.81·6-s − 6.84i·7-s + (−7.19 + 1.64i)8-s + (−6.77 + 8.49i)9-s + (−2.04 + 8.96i)10-s + (−8.76 + 10.9i)11-s + (5.48 − 11.3i)12-s + (1.36 + 5.99i)13-s + (6.65 + 3.20i)14-s + (−23.6 − 29.6i)15-s + (−0.749 + 3.28i)16-s + (−0.615 + 2.69i)17-s + ⋯
L(s)  = 1  + (−0.234 + 0.486i)2-s + (−0.644 − 1.33i)3-s + (0.441 + 0.553i)4-s + (1.66 − 0.379i)5-s + 0.802·6-s − 0.977i·7-s + (−0.899 + 0.205i)8-s + (−0.752 + 0.943i)9-s + (−0.204 + 0.896i)10-s + (−0.796 + 0.998i)11-s + (0.456 − 0.948i)12-s + (0.105 + 0.461i)13-s + (0.475 + 0.229i)14-s + (−1.57 − 1.97i)15-s + (−0.0468 + 0.205i)16-s + (−0.0361 + 0.158i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.940 + 0.339i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.940 + 0.339i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(43\)
Sign: $0.940 + 0.339i$
Analytic conductor: \(1.17166\)
Root analytic conductor: \(1.08243\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{43} (22, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 43,\ (\ :1),\ 0.940 + 0.339i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.02140 - 0.178646i\)
\(L(\frac12)\) \(\approx\) \(1.02140 - 0.178646i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad43 \( 1 + (-27.5 - 32.9i)T \)
good2 \( 1 + (0.468 - 0.972i)T + (-2.49 - 3.12i)T^{2} \)
3 \( 1 + (1.93 + 4.01i)T + (-5.61 + 7.03i)T^{2} \)
5 \( 1 + (-8.30 + 1.89i)T + (22.5 - 10.8i)T^{2} \)
7 \( 1 + 6.84iT - 49T^{2} \)
11 \( 1 + (8.76 - 10.9i)T + (-26.9 - 117. i)T^{2} \)
13 \( 1 + (-1.36 - 5.99i)T + (-152. + 73.3i)T^{2} \)
17 \( 1 + (0.615 - 2.69i)T + (-260. - 125. i)T^{2} \)
19 \( 1 + (-10.2 + 8.20i)T + (80.3 - 351. i)T^{2} \)
23 \( 1 + (11.5 - 14.4i)T + (-117. - 515. i)T^{2} \)
29 \( 1 + (22.3 - 46.4i)T + (-524. - 657. i)T^{2} \)
31 \( 1 + (10.3 + 4.98i)T + (599. + 751. i)T^{2} \)
37 \( 1 + 53.9iT - 1.36e3T^{2} \)
41 \( 1 + (-7.30 - 3.51i)T + (1.04e3 + 1.31e3i)T^{2} \)
47 \( 1 + (-18.4 - 23.1i)T + (-491. + 2.15e3i)T^{2} \)
53 \( 1 + (-7.84 + 34.3i)T + (-2.53e3 - 1.21e3i)T^{2} \)
59 \( 1 + (-23.1 + 101. i)T + (-3.13e3 - 1.51e3i)T^{2} \)
61 \( 1 + (33.6 + 69.8i)T + (-2.32e3 + 2.90e3i)T^{2} \)
67 \( 1 + (-1.19 - 1.50i)T + (-998. + 4.37e3i)T^{2} \)
71 \( 1 + (-6.33 + 5.04i)T + (1.12e3 - 4.91e3i)T^{2} \)
73 \( 1 + (59.4 - 13.5i)T + (4.80e3 - 2.31e3i)T^{2} \)
79 \( 1 - 22.0T + 6.24e3T^{2} \)
83 \( 1 + (37.5 - 18.0i)T + (4.29e3 - 5.38e3i)T^{2} \)
89 \( 1 + (7.92 + 16.4i)T + (-4.93e3 + 6.19e3i)T^{2} \)
97 \( 1 + (40.8 - 51.2i)T + (-2.09e3 - 9.17e3i)T^{2} \)
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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

−16.15968740902908168590319140176, −14.21069565102161167661250296775, −13.09686235531840888810529529562, −12.56418165544580438737984922896, −10.98196358998510663211103186936, −9.429762477537192802251714083038, −7.58313851369740876408873120340, −6.79928818948162487709636049062, −5.56739591002581868792388297704, −1.90194890860584164705421413696, 2.64701289545632065823527844308, 5.62659583927962498344181956486, 5.85325323788850399697011471660, 9.073935814079184421932554344306, 10.08269549639160372819203714416, 10.59854933443485939327455308619, 11.78349379435204450385464352392, 13.57961329126888253896685652376, 14.92653660411053384933095610108, 15.76102604921041929642125758887

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