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Properties

Label 2-43-43.37-c2-0-5
Degree $2$
Conductor $43$
Sign $-0.899 + 0.436i$
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  − 3.82i·2-s + (−0.868 − 0.501i)3-s − 10.6·4-s + (2.44 + 1.41i)5-s + (−1.91 + 3.31i)6-s + (9.14 − 5.27i)7-s + 25.2i·8-s + (−3.99 − 6.92i)9-s + (5.40 − 9.36i)10-s − 3.26·11-s + (9.21 + 5.32i)12-s + (0.795 + 1.37i)13-s + (−20.1 − 34.9i)14-s + (−1.41 − 2.45i)15-s + 54.2·16-s + (8.33 + 14.4i)17-s + ⋯
L(s)  = 1  − 1.91i·2-s + (−0.289 − 0.167i)3-s − 2.65·4-s + (0.489 + 0.282i)5-s + (−0.319 + 0.553i)6-s + (1.30 − 0.753i)7-s + 3.16i·8-s + (−0.444 − 0.769i)9-s + (0.540 − 0.936i)10-s − 0.296·11-s + (0.767 + 0.443i)12-s + (0.0612 + 0.106i)13-s + (−1.44 − 2.49i)14-s + (−0.0945 − 0.163i)15-s + 3.39·16-s + (0.490 + 0.849i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.899 + 0.436i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.899 + 0.436i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(43\)
Sign: $-0.899 + 0.436i$
Analytic conductor: \(1.17166\)
Root analytic conductor: \(1.08243\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{43} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 43,\ (\ :1),\ -0.899 + 0.436i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.218641 - 0.951745i\)
\(L(\frac12)\) \(\approx\) \(0.218641 - 0.951745i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad43 \( 1 + (-22.0 - 36.9i)T \)
good2 \( 1 + 3.82iT - 4T^{2} \)
3 \( 1 + (0.868 + 0.501i)T + (4.5 + 7.79i)T^{2} \)
5 \( 1 + (-2.44 - 1.41i)T + (12.5 + 21.6i)T^{2} \)
7 \( 1 + (-9.14 + 5.27i)T + (24.5 - 42.4i)T^{2} \)
11 \( 1 + 3.26T + 121T^{2} \)
13 \( 1 + (-0.795 - 1.37i)T + (-84.5 + 146. i)T^{2} \)
17 \( 1 + (-8.33 - 14.4i)T + (-144.5 + 250. i)T^{2} \)
19 \( 1 + (-31.3 - 18.0i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (0.918 - 1.59i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 + (25.9 - 14.9i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 + (2.48 - 4.31i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + (31.6 + 18.2i)T + (684.5 + 1.18e3i)T^{2} \)
41 \( 1 + 30.2T + 1.68e3T^{2} \)
47 \( 1 + 34.9T + 2.20e3T^{2} \)
53 \( 1 + (16.5 - 28.6i)T + (-1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + 6.28T + 3.48e3T^{2} \)
61 \( 1 + (-54.6 + 31.5i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-33.8 + 58.5i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + (-72.9 + 42.1i)T + (2.52e3 - 4.36e3i)T^{2} \)
73 \( 1 + (15.9 - 9.21i)T + (2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-43.3 - 75.0i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (6.92 - 11.9i)T + (-3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + (101. + 58.7i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + 53.6T + 9.40e3T^{2} \)
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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

−14.46236351264813582514944844301, −13.92964953444104948458912991509, −12.51205122479150716265477208434, −11.54853826795147319697275977461, −10.67391850321618427542708342026, −9.620107545871308225760908808573, −8.070799207275308050458521252719, −5.36389244298810356874137208481, −3.61948926353681548173324536116, −1.47402740923991551034019239154, 5.17446352873766703450904796930, 5.39545858272703767457748961752, 7.41036740231190815250582142081, 8.424795640386065899832269239054, 9.591542607048976898333090206025, 11.56601754856289674367183418983, 13.43342836586140127898170558041, 14.14861344489392844086449396626, 15.28193603138596614311762658326, 16.15127354994713094000549728055

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