Properties

Label 2-43-43.19-c6-0-1
Degree $2$
Conductor $43$
Sign $0.954 - 0.298i$
Analytic cond. $9.89232$
Root an. cond. $3.14520$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−6.36 − 5.07i)2-s + (−3.86 − 25.6i)3-s + (0.504 + 2.21i)4-s + (85.2 + 125. i)5-s + (−105. + 182. i)6-s + (−386. + 223. i)7-s + (−218. + 452. i)8-s + (53.1 − 16.3i)9-s + (92.0 − 1.22e3i)10-s + (244. − 1.06e3i)11-s + (54.8 − 21.5i)12-s + (308. + 4.12e3i)13-s + (3.59e3 + 541. i)14-s + (2.87e3 − 2.67e3i)15-s + (3.81e3 − 1.83e3i)16-s + (631. + 430. i)17-s + ⋯
L(s)  = 1  + (−0.795 − 0.634i)2-s + (−0.143 − 0.950i)3-s + (0.00789 + 0.0345i)4-s + (0.681 + 1.00i)5-s + (−0.489 + 0.846i)6-s + (−1.12 + 0.650i)7-s + (−0.425 + 0.884i)8-s + (0.0728 − 0.0224i)9-s + (0.0920 − 1.22i)10-s + (0.183 − 0.803i)11-s + (0.0317 − 0.0124i)12-s + (0.140 + 1.87i)13-s + (1.30 + 0.197i)14-s + (0.852 − 0.791i)15-s + (0.931 − 0.448i)16-s + (0.128 + 0.0875i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.954 - 0.298i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.954 - 0.298i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(43\)
Sign: $0.954 - 0.298i$
Analytic conductor: \(9.89232\)
Root analytic conductor: \(3.14520\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{43} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 43,\ (\ :3),\ 0.954 - 0.298i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.787390 + 0.120441i\)
\(L(\frac12)\) \(\approx\) \(0.787390 + 0.120441i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad43 \( 1 + (-4.89e4 - 6.26e4i)T \)
good2 \( 1 + (6.36 + 5.07i)T + (14.2 + 62.3i)T^{2} \)
3 \( 1 + (3.86 + 25.6i)T + (-696. + 214. i)T^{2} \)
5 \( 1 + (-85.2 - 125. i)T + (-5.70e3 + 1.45e4i)T^{2} \)
7 \( 1 + (386. - 223. i)T + (5.88e4 - 1.01e5i)T^{2} \)
11 \( 1 + (-244. + 1.06e3i)T + (-1.59e6 - 7.68e5i)T^{2} \)
13 \( 1 + (-308. - 4.12e3i)T + (-4.77e6 + 7.19e5i)T^{2} \)
17 \( 1 + (-631. - 430. i)T + (8.81e6 + 2.24e7i)T^{2} \)
19 \( 1 + (-556. + 1.80e3i)T + (-3.88e7 - 2.65e7i)T^{2} \)
23 \( 1 + (-6.11e3 - 5.67e3i)T + (1.10e7 + 1.47e8i)T^{2} \)
29 \( 1 + (1.60e3 - 1.06e4i)T + (-5.68e8 - 1.75e8i)T^{2} \)
31 \( 1 + (-1.07e4 - 2.75e4i)T + (-6.50e8 + 6.03e8i)T^{2} \)
37 \( 1 + (4.49e4 + 2.59e4i)T + (1.28e9 + 2.22e9i)T^{2} \)
41 \( 1 + (6.73e4 - 8.44e4i)T + (-1.05e9 - 4.63e9i)T^{2} \)
47 \( 1 + (-2.79e4 - 1.22e5i)T + (-9.71e9 + 4.67e9i)T^{2} \)
53 \( 1 + (1.20e4 - 1.61e5i)T + (-2.19e10 - 3.30e9i)T^{2} \)
59 \( 1 + (1.83e4 - 8.85e3i)T + (2.62e10 - 3.29e10i)T^{2} \)
61 \( 1 + (-3.73e5 - 1.46e5i)T + (3.77e10 + 3.50e10i)T^{2} \)
67 \( 1 + (-3.38e5 - 1.04e5i)T + (7.47e10 + 5.09e10i)T^{2} \)
71 \( 1 + (4.21e5 + 4.54e5i)T + (-9.57e9 + 1.27e11i)T^{2} \)
73 \( 1 + (-4.09e4 + 3.07e3i)T + (1.49e11 - 2.25e10i)T^{2} \)
79 \( 1 + (2.96e5 + 5.13e5i)T + (-1.21e11 + 2.10e11i)T^{2} \)
83 \( 1 + (6.19e5 - 9.33e4i)T + (3.12e11 - 9.63e10i)T^{2} \)
89 \( 1 + (-8.79e4 - 5.83e5i)T + (-4.74e11 + 1.46e11i)T^{2} \)
97 \( 1 + (2.00e4 - 8.79e4i)T + (-7.50e11 - 3.61e11i)T^{2} \)
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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

−14.44885746871010767776686321198, −13.54808501566348259593276673575, −12.14291908292481297449758830220, −11.09919232951816564820386401482, −9.796144242011171830338190173015, −8.925074479748510107500966685574, −6.84472427304849950469367325300, −6.08133931245553559459783227053, −2.82121811054856500657294965379, −1.47034980566060100028877907263, 0.52748272911308297092870377003, 3.71957090678027311533829121781, 5.40301624246279601075002663936, 7.07135614625198527023192943817, 8.582388017797621793021111536217, 9.889183713861916044081276137780, 10.10311142311875735483955563255, 12.62619838070839383441101506470, 13.20690168559612485536555291508, 15.31012599338179018392038641864

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