Properties

Label 2-43-43.38-c1-0-0
Degree $2$
Conductor $43$
Sign $0.783 - 0.621i$
Analytic cond. $0.343356$
Root an. cond. $0.585966$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.178 + 0.780i)2-s + (−0.711 − 0.219i)3-s + (1.22 + 0.589i)4-s + (0.511 + 1.30i)5-s + (0.298 − 0.516i)6-s + (−2.37 − 4.12i)7-s + (−1.67 + 2.10i)8-s + (−2.02 − 1.37i)9-s + (−1.10 + 0.167i)10-s + (2.39 − 1.15i)11-s + (−0.741 − 0.688i)12-s + (0.611 + 0.0920i)13-s + (3.64 − 1.12i)14-s + (−0.0779 − 1.04i)15-s + (0.352 + 0.441i)16-s + (−2.15 + 5.50i)17-s + ⋯
L(s)  = 1  + (−0.125 + 0.551i)2-s + (−0.410 − 0.126i)3-s + (0.612 + 0.294i)4-s + (0.228 + 0.583i)5-s + (0.121 − 0.210i)6-s + (−0.899 − 1.55i)7-s + (−0.592 + 0.743i)8-s + (−0.673 − 0.459i)9-s + (−0.350 + 0.0528i)10-s + (0.722 − 0.348i)11-s + (−0.214 − 0.198i)12-s + (0.169 + 0.0255i)13-s + (0.972 − 0.300i)14-s + (−0.0201 − 0.268i)15-s + (0.0880 + 0.110i)16-s + (−0.523 + 1.33i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.783 - 0.621i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.783 - 0.621i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(43\)
Sign: $0.783 - 0.621i$
Analytic conductor: \(0.343356\)
Root analytic conductor: \(0.585966\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{43} (38, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 43,\ (\ :1/2),\ 0.783 - 0.621i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.685458 + 0.238697i\)
\(L(\frac12)\) \(\approx\) \(0.685458 + 0.238697i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad43 \( 1 + (6.54 + 0.452i)T \)
good2 \( 1 + (0.178 - 0.780i)T + (-1.80 - 0.867i)T^{2} \)
3 \( 1 + (0.711 + 0.219i)T + (2.47 + 1.68i)T^{2} \)
5 \( 1 + (-0.511 - 1.30i)T + (-3.66 + 3.40i)T^{2} \)
7 \( 1 + (2.37 + 4.12i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2.39 + 1.15i)T + (6.85 - 8.60i)T^{2} \)
13 \( 1 + (-0.611 - 0.0920i)T + (12.4 + 3.83i)T^{2} \)
17 \( 1 + (2.15 - 5.50i)T + (-12.4 - 11.5i)T^{2} \)
19 \( 1 + (1.48 - 1.01i)T + (6.94 - 17.6i)T^{2} \)
23 \( 1 + (-0.178 + 2.37i)T + (-22.7 - 3.42i)T^{2} \)
29 \( 1 + (1.79 - 0.555i)T + (23.9 - 16.3i)T^{2} \)
31 \( 1 + (-5.49 - 5.09i)T + (2.31 + 30.9i)T^{2} \)
37 \( 1 + (-1.11 + 1.92i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (0.583 - 2.55i)T + (-36.9 - 17.7i)T^{2} \)
47 \( 1 + (8.07 + 3.88i)T + (29.3 + 36.7i)T^{2} \)
53 \( 1 + (-13.5 + 2.04i)T + (50.6 - 15.6i)T^{2} \)
59 \( 1 + (4.23 + 5.30i)T + (-13.1 + 57.5i)T^{2} \)
61 \( 1 + (-3.03 + 2.81i)T + (4.55 - 60.8i)T^{2} \)
67 \( 1 + (-4.65 + 3.17i)T + (24.4 - 62.3i)T^{2} \)
71 \( 1 + (0.641 + 8.55i)T + (-70.2 + 10.5i)T^{2} \)
73 \( 1 + (-8.41 - 1.26i)T + (69.7 + 21.5i)T^{2} \)
79 \( 1 + (-3.09 - 5.36i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (1.21 + 0.374i)T + (68.5 + 46.7i)T^{2} \)
89 \( 1 + (9.04 + 2.79i)T + (73.5 + 50.1i)T^{2} \)
97 \( 1 + (-4.06 + 1.95i)T + (60.4 - 75.8i)T^{2} \)
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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

−16.55207144978960255227524872223, −15.02267759101327636140706981338, −14.02441639465546991598716973660, −12.62413883811122254554828393666, −11.23537136076974712007017723714, −10.32081307579905870285247966831, −8.452370757451733455913983724659, −6.69043799420212312690847975563, −6.40849836052162299778787954914, −3.50821162799008248289469189440, 2.60514349455380226912408488264, 5.39183532714152129797005776017, 6.53610386599032064408175257162, 8.902409583769967728086372882141, 9.755061056819501252047637978047, 11.43002280173021349688388947783, 12.00008692563994333967588770877, 13.24519132386054470830692184896, 15.03088912992116517147559937373, 15.92079963806934145810443742270

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