Properties

Label 2-43-43.4-c3-0-7
Degree $2$
Conductor $43$
Sign $-0.957 + 0.287i$
Analytic cond. $2.53708$
Root an. cond. $1.59282$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.976 + 1.22i)2-s + (−3.38 − 4.24i)3-s + (1.23 + 5.40i)4-s + (−7.32 − 3.52i)5-s + 8.49·6-s − 30.7·7-s + (−19.1 − 9.20i)8-s + (−0.538 + 2.36i)9-s + (11.4 − 5.52i)10-s + (−2.01 + 8.83i)11-s + (18.7 − 23.5i)12-s + (14.5 + 6.98i)13-s + (30.0 − 37.6i)14-s + (9.80 + 42.9i)15-s + (−10.0 + 4.83i)16-s + (12.0 − 5.82i)17-s + ⋯
L(s)  = 1  + (−0.345 + 0.432i)2-s + (−0.650 − 0.816i)3-s + (0.154 + 0.675i)4-s + (−0.654 − 0.315i)5-s + 0.578·6-s − 1.65·7-s + (−0.844 − 0.406i)8-s + (−0.0199 + 0.0874i)9-s + (0.362 − 0.174i)10-s + (−0.0552 + 0.242i)11-s + (0.451 − 0.565i)12-s + (0.309 + 0.149i)13-s + (0.572 − 0.718i)14-s + (0.168 + 0.739i)15-s + (−0.156 + 0.0755i)16-s + (0.172 − 0.0831i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.957 + 0.287i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.957 + 0.287i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(43\)
Sign: $-0.957 + 0.287i$
Analytic conductor: \(2.53708\)
Root analytic conductor: \(1.59282\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{43} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 43,\ (\ :3/2),\ -0.957 + 0.287i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.00724330 - 0.0493992i\)
\(L(\frac12)\) \(\approx\) \(0.00724330 - 0.0493992i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad43 \( 1 + (279. - 39.2i)T \)
good2 \( 1 + (0.976 - 1.22i)T + (-1.78 - 7.79i)T^{2} \)
3 \( 1 + (3.38 + 4.24i)T + (-6.00 + 26.3i)T^{2} \)
5 \( 1 + (7.32 + 3.52i)T + (77.9 + 97.7i)T^{2} \)
7 \( 1 + 30.7T + 343T^{2} \)
11 \( 1 + (2.01 - 8.83i)T + (-1.19e3 - 577. i)T^{2} \)
13 \( 1 + (-14.5 - 6.98i)T + (1.36e3 + 1.71e3i)T^{2} \)
17 \( 1 + (-12.0 + 5.82i)T + (3.06e3 - 3.84e3i)T^{2} \)
19 \( 1 + (-17.0 - 74.7i)T + (-6.17e3 + 2.97e3i)T^{2} \)
23 \( 1 + (-8.55 + 37.4i)T + (-1.09e4 - 5.27e3i)T^{2} \)
29 \( 1 + (-156. + 196. i)T + (-5.42e3 - 2.37e4i)T^{2} \)
31 \( 1 + (75.2 - 94.4i)T + (-6.62e3 - 2.90e4i)T^{2} \)
37 \( 1 + 391.T + 5.06e4T^{2} \)
41 \( 1 + (211. - 264. i)T + (-1.53e4 - 6.71e4i)T^{2} \)
47 \( 1 + (26.7 + 117. i)T + (-9.35e4 + 4.50e4i)T^{2} \)
53 \( 1 + (-433. + 208. i)T + (9.28e4 - 1.16e5i)T^{2} \)
59 \( 1 + (322. - 155. i)T + (1.28e5 - 1.60e5i)T^{2} \)
61 \( 1 + (369. + 463. i)T + (-5.05e4 + 2.21e5i)T^{2} \)
67 \( 1 + (-149. - 655. i)T + (-2.70e5 + 1.30e5i)T^{2} \)
71 \( 1 + (93.0 + 407. i)T + (-3.22e5 + 1.55e5i)T^{2} \)
73 \( 1 + (-831. - 400. i)T + (2.42e5 + 3.04e5i)T^{2} \)
79 \( 1 + 107.T + 4.93e5T^{2} \)
83 \( 1 + (625. + 784. i)T + (-1.27e5 + 5.57e5i)T^{2} \)
89 \( 1 + (117. + 147. i)T + (-1.56e5 + 6.87e5i)T^{2} \)
97 \( 1 + (107. - 469. i)T + (-8.22e5 - 3.95e5i)T^{2} \)
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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

−15.48468581668417735062599004778, −13.37270266099706413270908646421, −12.35425705758004416410020428662, −11.92947475079383241193909707056, −9.871430220145332561055245097390, −8.386149095160429968169633445221, −7.04733605680915742926209147663, −6.23264215299408084551911064429, −3.54286736861492852134174783599, −0.04603071787579590910541255792, 3.36004993566397471027153350886, 5.42710626364945978412033643730, 6.81172816591266449063452318104, 9.096214332898880510816060958237, 10.19212931145296111728397356842, 10.91119316584577376481195812361, 12.07510081704742677799612626995, 13.63271291768999985096244088623, 15.43589855162024387296221239883, 15.75238657290183754272536722507

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