Properties

Degree $2$
Conductor $43$
Sign $0.989 - 0.143i$
Motivic weight $7$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−11.9 − 14.9i)2-s + (22.8 + 3.44i)3-s + (−53.3 + 233. i)4-s + (−375. − 255. i)5-s + (−221. − 383. i)6-s + (97.1 − 168. i)7-s + (1.93e3 − 930. i)8-s + (−1.58e3 − 487. i)9-s + (650. + 8.67e3i)10-s + (341. + 1.49e3i)11-s + (−2.02e3 + 5.15e3i)12-s + (−530. + 7.07e3i)13-s + (−3.68e3 + 555. i)14-s + (−7.68e3 − 7.12e3i)15-s + (−9.38e3 − 4.52e3i)16-s + (−1.40e3 + 956. i)17-s + ⋯
L(s)  = 1  + (−1.05 − 1.32i)2-s + (0.488 + 0.0735i)3-s + (−0.416 + 1.82i)4-s + (−1.34 − 0.914i)5-s + (−0.418 − 0.724i)6-s + (0.107 − 0.185i)7-s + (1.33 − 0.642i)8-s + (−0.722 − 0.222i)9-s + (0.205 + 2.74i)10-s + (0.0773 + 0.338i)11-s + (−0.337 + 0.861i)12-s + (−0.0669 + 0.893i)13-s + (−0.358 + 0.0540i)14-s + (−0.587 − 0.545i)15-s + (−0.572 − 0.275i)16-s + (−0.0692 + 0.0472i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.989 - 0.143i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.989 - 0.143i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(43\)
Sign: $0.989 - 0.143i$
Motivic weight: \(7\)
Character: $\chi_{43} (9, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 43,\ (\ :7/2),\ 0.989 - 0.143i)\)

Particular Values

\(L(4)\) \(\approx\) \(0.400832 + 0.0289510i\)
\(L(\frac12)\) \(\approx\) \(0.400832 + 0.0289510i\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad43 \( 1 + (-1.10e5 - 5.09e5i)T \)
good2 \( 1 + (11.9 + 14.9i)T + (-28.4 + 124. i)T^{2} \)
3 \( 1 + (-22.8 - 3.44i)T + (2.08e3 + 644. i)T^{2} \)
5 \( 1 + (375. + 255. i)T + (2.85e4 + 7.27e4i)T^{2} \)
7 \( 1 + (-97.1 + 168. i)T + (-4.11e5 - 7.13e5i)T^{2} \)
11 \( 1 + (-341. - 1.49e3i)T + (-1.75e7 + 8.45e6i)T^{2} \)
13 \( 1 + (530. - 7.07e3i)T + (-6.20e7 - 9.35e6i)T^{2} \)
17 \( 1 + (1.40e3 - 956. i)T + (1.49e8 - 3.81e8i)T^{2} \)
19 \( 1 + (-2.76e4 + 8.54e3i)T + (7.38e8 - 5.03e8i)T^{2} \)
23 \( 1 + (-3.81e4 + 3.54e4i)T + (2.54e8 - 3.39e9i)T^{2} \)
29 \( 1 + (-6.79e4 + 1.02e4i)T + (1.64e10 - 5.08e9i)T^{2} \)
31 \( 1 + (8.76e4 - 2.23e5i)T + (-2.01e10 - 1.87e10i)T^{2} \)
37 \( 1 + (1.58e5 + 2.75e5i)T + (-4.74e10 + 8.22e10i)T^{2} \)
41 \( 1 + (-4.16e5 - 5.22e5i)T + (-4.33e10 + 1.89e11i)T^{2} \)
47 \( 1 + (9.35e4 - 4.09e5i)T + (-4.56e11 - 2.19e11i)T^{2} \)
53 \( 1 + (-2.88e3 - 3.85e4i)T + (-1.16e12 + 1.75e11i)T^{2} \)
59 \( 1 + (1.77e6 + 8.52e5i)T + (1.55e12 + 1.94e12i)T^{2} \)
61 \( 1 + (-1.85e5 - 4.72e5i)T + (-2.30e12 + 2.13e12i)T^{2} \)
67 \( 1 + (2.98e6 - 9.21e5i)T + (5.00e12 - 3.41e12i)T^{2} \)
71 \( 1 + (2.66e6 + 2.47e6i)T + (6.79e11 + 9.06e12i)T^{2} \)
73 \( 1 + (3.12e5 - 4.17e6i)T + (-1.09e13 - 1.64e12i)T^{2} \)
79 \( 1 + (2.34e6 - 4.05e6i)T + (-9.60e12 - 1.66e13i)T^{2} \)
83 \( 1 + (-4.52e6 - 6.81e5i)T + (2.59e13 + 7.99e12i)T^{2} \)
89 \( 1 + (1.01e7 + 1.53e6i)T + (4.22e13 + 1.30e13i)T^{2} \)
97 \( 1 + (2.58e6 + 1.13e7i)T + (-7.27e13 + 3.50e13i)T^{2} \)
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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

−14.38948916874896535590663008220, −12.68634895823904518680700704786, −11.84433259875518027463996802782, −11.02925318831778261739432119565, −9.325711480566701827597344904283, −8.672229261023704590383652773170, −7.58265326139459646058133240304, −4.39052337530609400022967099151, −3.03545385482681397691511009416, −1.11985484698229386956788155409, 0.27281965271600650263971339994, 3.22624168153385766995940154667, 5.66215869553769670961741575636, 7.27637582651239516380317096019, 7.916888135939419204393434978530, 8.942816266052101554043540581947, 10.52861285119802041312234960357, 11.73490703359611869630379083844, 13.88929850093006607248410190551, 14.99392487013583036815325893353

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