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Properties

Label 2-43-43.41-c1-0-0
Degree $2$
Conductor $43$
Sign $0.397 - 0.917i$
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  + (−1.62 + 0.781i)2-s + (1.12 + 0.541i)3-s + (0.777 − 0.974i)4-s + (−0.5 + 2.19i)5-s − 2.24·6-s + 1.69·7-s + (0.301 − 1.32i)8-s + (−0.900 − 1.12i)9-s + (−0.900 − 3.94i)10-s + (−0.708 − 0.888i)11-s + (1.40 − 0.674i)12-s + (1.45 − 6.37i)13-s + (−2.74 + 1.32i)14-s + (−1.74 + 2.19i)15-s + (1.09 + 4.81i)16-s + (0.801 + 3.51i)17-s + ⋯
L(s)  = 1  + (−1.14 + 0.552i)2-s + (0.648 + 0.312i)3-s + (0.388 − 0.487i)4-s + (−0.223 + 0.979i)5-s − 0.917·6-s + 0.639·7-s + (0.106 − 0.467i)8-s + (−0.300 − 0.376i)9-s + (−0.284 − 1.24i)10-s + (−0.213 − 0.268i)11-s + (0.404 − 0.194i)12-s + (0.403 − 1.76i)13-s + (−0.734 + 0.353i)14-s + (−0.451 + 0.565i)15-s + (0.274 + 1.20i)16-s + (0.194 + 0.852i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.397 - 0.917i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.397 - 0.917i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(43\)
Sign: $0.397 - 0.917i$
Analytic conductor: \(0.343356\)
Root analytic conductor: \(0.585966\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{43} (41, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 43,\ (\ :1/2),\ 0.397 - 0.917i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.462736 + 0.303702i\)
\(L(\frac12)\) \(\approx\) \(0.462736 + 0.303702i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad43 \( 1 + (-1.67 - 6.34i)T \)
good2 \( 1 + (1.62 - 0.781i)T + (1.24 - 1.56i)T^{2} \)
3 \( 1 + (-1.12 - 0.541i)T + (1.87 + 2.34i)T^{2} \)
5 \( 1 + (0.5 - 2.19i)T + (-4.50 - 2.16i)T^{2} \)
7 \( 1 - 1.69T + 7T^{2} \)
11 \( 1 + (0.708 + 0.888i)T + (-2.44 + 10.7i)T^{2} \)
13 \( 1 + (-1.45 + 6.37i)T + (-11.7 - 5.64i)T^{2} \)
17 \( 1 + (-0.801 - 3.51i)T + (-15.3 + 7.37i)T^{2} \)
19 \( 1 + (2.77 - 3.47i)T + (-4.22 - 18.5i)T^{2} \)
23 \( 1 + (2.31 + 2.90i)T + (-5.11 + 22.4i)T^{2} \)
29 \( 1 + (-5.40 + 2.60i)T + (18.0 - 22.6i)T^{2} \)
31 \( 1 + (9.30 - 4.47i)T + (19.3 - 24.2i)T^{2} \)
37 \( 1 - 3.65T + 37T^{2} \)
41 \( 1 + (3.96 - 1.90i)T + (25.5 - 32.0i)T^{2} \)
47 \( 1 + (1.96 - 2.46i)T + (-10.4 - 45.8i)T^{2} \)
53 \( 1 + (2.48 + 10.8i)T + (-47.7 + 22.9i)T^{2} \)
59 \( 1 + (0.0353 + 0.154i)T + (-53.1 + 25.5i)T^{2} \)
61 \( 1 + (-0.643 - 0.309i)T + (38.0 + 47.6i)T^{2} \)
67 \( 1 + (1.30 - 1.64i)T + (-14.9 - 65.3i)T^{2} \)
71 \( 1 + (2.32 - 2.91i)T + (-15.7 - 69.2i)T^{2} \)
73 \( 1 + (0.318 - 1.39i)T + (-65.7 - 31.6i)T^{2} \)
79 \( 1 + 0.0609T + 79T^{2} \)
83 \( 1 + (-11.0 - 5.32i)T + (51.7 + 64.8i)T^{2} \)
89 \( 1 + (-9.56 - 4.60i)T + (55.4 + 69.5i)T^{2} \)
97 \( 1 + (-2.18 - 2.73i)T + (-21.5 + 94.5i)T^{2} \)
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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

−16.21761284789774000606856224542, −15.05646967789241366184322902372, −14.55606461491704760005382989649, −12.78866234980566871294198577463, −10.87240174158015935073076615029, −10.07168684269815616432323775524, −8.441261838996480769431665392131, −7.934675546452627457963937252252, −6.23530673000768311636743758546, −3.44152526760989267585573380952, 1.93492967891921699407870426510, 4.84318037718444795871189823885, 7.52255118476162318472367986834, 8.683375405109862792957169873101, 9.239100655431145950324598655377, 10.98185614852348046052426302624, 11.92629051742724861714394941611, 13.52677134669434850024111277010, 14.45941066653563347649235953629, 16.23528004159957939257690855250

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