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Properties

Label 2-43-43.22-c2-0-5
Degree $2$
Conductor $43$
Sign $0.322 + 0.946i$
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  + (1.68 − 3.50i)2-s + (2.07 + 4.30i)3-s + (−6.93 − 8.70i)4-s + (−2.02 + 0.462i)5-s + 18.5·6-s + 2.90i·7-s + (−27.0 + 6.16i)8-s + (−8.61 + 10.7i)9-s + (−1.79 + 7.88i)10-s + (2.92 − 3.66i)11-s + (23.0 − 47.8i)12-s + (0.850 + 3.72i)13-s + (10.1 + 4.90i)14-s + (−6.19 − 7.76i)15-s + (−14.0 + 61.7i)16-s + (3.19 − 13.9i)17-s + ⋯
L(s)  = 1  + (0.843 − 1.75i)2-s + (0.690 + 1.43i)3-s + (−1.73 − 2.17i)4-s + (−0.405 + 0.0925i)5-s + 3.09·6-s + 0.415i·7-s + (−3.37 + 0.771i)8-s + (−0.956 + 1.19i)9-s + (−0.179 + 0.788i)10-s + (0.265 − 0.333i)11-s + (1.92 − 3.99i)12-s + (0.0653 + 0.286i)13-s + (0.728 + 0.350i)14-s + (−0.412 − 0.517i)15-s + (−0.880 + 3.85i)16-s + (0.187 − 0.822i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.322 + 0.946i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.322 + 0.946i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(43\)
Sign: $0.322 + 0.946i$
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: \((2,\ 43,\ (\ :1),\ 0.322 + 0.946i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.29671 - 0.928421i\)
\(L(\frac12)\) \(\approx\) \(1.29671 - 0.928421i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad43 \( 1 + (8.53 - 42.1i)T \)
good2 \( 1 + (-1.68 + 3.50i)T + (-2.49 - 3.12i)T^{2} \)
3 \( 1 + (-2.07 - 4.30i)T + (-5.61 + 7.03i)T^{2} \)
5 \( 1 + (2.02 - 0.462i)T + (22.5 - 10.8i)T^{2} \)
7 \( 1 - 2.90iT - 49T^{2} \)
11 \( 1 + (-2.92 + 3.66i)T + (-26.9 - 117. i)T^{2} \)
13 \( 1 + (-0.850 - 3.72i)T + (-152. + 73.3i)T^{2} \)
17 \( 1 + (-3.19 + 13.9i)T + (-260. - 125. i)T^{2} \)
19 \( 1 + (-9.17 + 7.32i)T + (80.3 - 351. i)T^{2} \)
23 \( 1 + (-18.1 + 22.7i)T + (-117. - 515. i)T^{2} \)
29 \( 1 + (10.9 - 22.6i)T + (-524. - 657. i)T^{2} \)
31 \( 1 + (43.5 + 20.9i)T + (599. + 751. i)T^{2} \)
37 \( 1 - 10.6iT - 1.36e3T^{2} \)
41 \( 1 + (-44.4 - 21.3i)T + (1.04e3 + 1.31e3i)T^{2} \)
47 \( 1 + (3.61 + 4.53i)T + (-491. + 2.15e3i)T^{2} \)
53 \( 1 + (-16.1 + 70.8i)T + (-2.53e3 - 1.21e3i)T^{2} \)
59 \( 1 + (1.88 - 8.25i)T + (-3.13e3 - 1.51e3i)T^{2} \)
61 \( 1 + (-7.04 - 14.6i)T + (-2.32e3 + 2.90e3i)T^{2} \)
67 \( 1 + (-17.4 - 21.8i)T + (-998. + 4.37e3i)T^{2} \)
71 \( 1 + (10.3 - 8.28i)T + (1.12e3 - 4.91e3i)T^{2} \)
73 \( 1 + (46.0 - 10.5i)T + (4.80e3 - 2.31e3i)T^{2} \)
79 \( 1 + 133.T + 6.24e3T^{2} \)
83 \( 1 + (-85.8 + 41.3i)T + (4.29e3 - 5.38e3i)T^{2} \)
89 \( 1 + (-65.7 - 136. i)T + (-4.93e3 + 6.19e3i)T^{2} \)
97 \( 1 + (23.2 - 29.1i)T + (-2.09e3 - 9.17e3i)T^{2} \)
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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

−14.95288598746757139770658711996, −14.37387659313420493497755508424, −13.12898446137516348693050500635, −11.65589244043349039028724172551, −10.89230423706516645034056885684, −9.632744069552650269418205403730, −8.933763755701199054493213752953, −5.25400308805773096847992102177, −4.02908994881223342234279239351, −2.88585374609948948173600763190, 3.74458616555903821682996896674, 5.82780129744774681271941018553, 7.23720197827321314448651175251, 7.73125603852797276509002867290, 8.943525980282621811655699625717, 12.12042245707302893355130768675, 12.96429389043027519428718973226, 13.80913288833297210012296928418, 14.65697555816427716219997290481, 15.67608835850473640908075010153

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