Properties

Label 2-43-43.40-c1-0-0
Degree $2$
Conductor $43$
Sign $0.506 - 0.862i$
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  + (−2.05 − 0.989i)2-s + (−0.239 + 3.20i)3-s + (1.99 + 2.49i)4-s + (−0.131 − 0.0405i)5-s + (3.65 − 6.33i)6-s + (0.934 + 1.61i)7-s + (−0.607 − 2.66i)8-s + (−7.22 − 1.08i)9-s + (0.230 + 0.213i)10-s + (1.63 − 2.05i)11-s + (−8.47 + 5.78i)12-s + (1.27 − 1.18i)13-s + (−0.318 − 4.24i)14-s + (0.161 − 0.411i)15-s + (0.0382 − 0.167i)16-s + (1.45 − 0.448i)17-s + ⋯
L(s)  = 1  + (−1.45 − 0.699i)2-s + (−0.138 + 1.84i)3-s + (0.996 + 1.24i)4-s + (−0.0588 − 0.0181i)5-s + (1.49 − 2.58i)6-s + (0.353 + 0.611i)7-s + (−0.214 − 0.940i)8-s + (−2.40 − 0.362i)9-s + (0.0727 + 0.0675i)10-s + (0.493 − 0.618i)11-s + (−2.44 + 1.66i)12-s + (0.354 − 0.329i)13-s + (−0.0851 − 1.13i)14-s + (0.0416 − 0.106i)15-s + (0.00956 − 0.0418i)16-s + (0.352 − 0.108i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.506 - 0.862i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.506 - 0.862i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(43\)
Sign: $0.506 - 0.862i$
Analytic conductor: \(0.343356\)
Root analytic conductor: \(0.585966\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{43} (40, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 43,\ (\ :1/2),\ 0.506 - 0.862i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.360327 + 0.206121i\)
\(L(\frac12)\) \(\approx\) \(0.360327 + 0.206121i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad43 \( 1 + (5.73 + 3.17i)T \)
good2 \( 1 + (2.05 + 0.989i)T + (1.24 + 1.56i)T^{2} \)
3 \( 1 + (0.239 - 3.20i)T + (-2.96 - 0.447i)T^{2} \)
5 \( 1 + (0.131 + 0.0405i)T + (4.13 + 2.81i)T^{2} \)
7 \( 1 + (-0.934 - 1.61i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.63 + 2.05i)T + (-2.44 - 10.7i)T^{2} \)
13 \( 1 + (-1.27 + 1.18i)T + (0.971 - 12.9i)T^{2} \)
17 \( 1 + (-1.45 + 0.448i)T + (14.0 - 9.57i)T^{2} \)
19 \( 1 + (-4.83 + 0.729i)T + (18.1 - 5.60i)T^{2} \)
23 \( 1 + (-1.59 - 4.07i)T + (-16.8 + 15.6i)T^{2} \)
29 \( 1 + (0.00236 + 0.0315i)T + (-28.6 + 4.32i)T^{2} \)
31 \( 1 + (2.33 - 1.59i)T + (11.3 - 28.8i)T^{2} \)
37 \( 1 + (0.00465 - 0.00806i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-7.49 - 3.61i)T + (25.5 + 32.0i)T^{2} \)
47 \( 1 + (3.50 + 4.39i)T + (-10.4 + 45.8i)T^{2} \)
53 \( 1 + (5.67 + 5.26i)T + (3.96 + 52.8i)T^{2} \)
59 \( 1 + (-0.483 + 2.11i)T + (-53.1 - 25.5i)T^{2} \)
61 \( 1 + (10.0 + 6.84i)T + (22.2 + 56.7i)T^{2} \)
67 \( 1 + (-7.65 + 1.15i)T + (64.0 - 19.7i)T^{2} \)
71 \( 1 + (4.78 - 12.2i)T + (-52.0 - 48.2i)T^{2} \)
73 \( 1 + (3.48 - 3.23i)T + (5.45 - 72.7i)T^{2} \)
79 \( 1 + (-6.00 - 10.3i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-0.321 + 4.29i)T + (-82.0 - 12.3i)T^{2} \)
89 \( 1 + (-0.574 + 7.67i)T + (-88.0 - 13.2i)T^{2} \)
97 \( 1 + (-1.00 + 1.25i)T + (-21.5 - 94.5i)T^{2} \)
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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

−16.32904696285889842491104749552, −15.57159067945851118877138926143, −14.25362636098403677717510447851, −11.72211243186263424196754833425, −11.16937337202587608915302068386, −9.971152835962436105565380203016, −9.217019443983061254853722899201, −8.224105479911207641282160473232, −5.41391343013823291794628843634, −3.34038933297299236333707036314, 1.38662562055602725834297614468, 6.16171255886666401370558536329, 7.28308176758742857696238175657, 7.87516351488385588688172889259, 9.260344280570924187851967948212, 10.98004594721995255770835600602, 12.20453752576703899416275023649, 13.56524838128734758044366035832, 14.68475912521443201172904342133, 16.41135431402945268353169216873

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