Properties

Label 2-43-43.24-c7-0-8
Degree $2$
Conductor $43$
Sign $0.374 - 0.927i$
Analytic cond. $13.4325$
Root an. cond. $3.66504$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (7.72 − 9.68i)2-s + (15.1 − 2.27i)3-s + (−5.66 − 24.8i)4-s + (−308. + 210. i)5-s + (94.7 − 164. i)6-s + (75.0 + 130. i)7-s + (1.14e3 + 551. i)8-s + (−1.86e3 + 575. i)9-s + (−345. + 4.61e3i)10-s + (−1.58e3 + 6.96e3i)11-s + (−142. − 362. i)12-s + (−139. − 1.85e3i)13-s + (1.83e3 + 277. i)14-s + (−4.18e3 + 3.88e3i)15-s + (1.71e4 − 8.24e3i)16-s + (1.59e3 + 1.08e3i)17-s + ⋯
L(s)  = 1  + (0.682 − 0.856i)2-s + (0.323 − 0.0487i)3-s + (−0.0442 − 0.193i)4-s + (−1.10 + 0.752i)5-s + (0.179 − 0.310i)6-s + (0.0827 + 0.143i)7-s + (0.790 + 0.380i)8-s + (−0.853 + 0.263i)9-s + (−0.109 + 1.45i)10-s + (−0.359 + 1.57i)11-s + (−0.0237 − 0.0605i)12-s + (−0.0175 − 0.234i)13-s + (0.179 + 0.0270i)14-s + (−0.320 + 0.297i)15-s + (1.04 − 0.502i)16-s + (0.0785 + 0.0535i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.374 - 0.927i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.374 - 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(43\)
Sign: $0.374 - 0.927i$
Analytic conductor: \(13.4325\)
Root analytic conductor: \(3.66504\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{43} (24, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 43,\ (\ :7/2),\ 0.374 - 0.927i)\)

Particular Values

\(L(4)\) \(\approx\) \(1.46364 + 0.987635i\)
\(L(\frac12)\) \(\approx\) \(1.46364 + 0.987635i\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad43 \( 1 + (4.68e5 + 2.27e5i)T \)
good2 \( 1 + (-7.72 + 9.68i)T + (-28.4 - 124. i)T^{2} \)
3 \( 1 + (-15.1 + 2.27i)T + (2.08e3 - 644. i)T^{2} \)
5 \( 1 + (308. - 210. i)T + (2.85e4 - 7.27e4i)T^{2} \)
7 \( 1 + (-75.0 - 130. i)T + (-4.11e5 + 7.13e5i)T^{2} \)
11 \( 1 + (1.58e3 - 6.96e3i)T + (-1.75e7 - 8.45e6i)T^{2} \)
13 \( 1 + (139. + 1.85e3i)T + (-6.20e7 + 9.35e6i)T^{2} \)
17 \( 1 + (-1.59e3 - 1.08e3i)T + (1.49e8 + 3.81e8i)T^{2} \)
19 \( 1 + (1.16e4 + 3.59e3i)T + (7.38e8 + 5.03e8i)T^{2} \)
23 \( 1 + (-2.14e4 - 1.98e4i)T + (2.54e8 + 3.39e9i)T^{2} \)
29 \( 1 + (-1.25e5 - 1.88e4i)T + (1.64e10 + 5.08e9i)T^{2} \)
31 \( 1 + (1.40e4 + 3.58e4i)T + (-2.01e10 + 1.87e10i)T^{2} \)
37 \( 1 + (-4.81e4 + 8.34e4i)T + (-4.74e10 - 8.22e10i)T^{2} \)
41 \( 1 + (4.28e5 - 5.36e5i)T + (-4.33e10 - 1.89e11i)T^{2} \)
47 \( 1 + (5.00e4 + 2.19e5i)T + (-4.56e11 + 2.19e11i)T^{2} \)
53 \( 1 + (-1.76e4 + 2.34e5i)T + (-1.16e12 - 1.75e11i)T^{2} \)
59 \( 1 + (-8.44e5 + 4.06e5i)T + (1.55e12 - 1.94e12i)T^{2} \)
61 \( 1 + (1.09e6 - 2.79e6i)T + (-2.30e12 - 2.13e12i)T^{2} \)
67 \( 1 + (1.62e6 + 5.02e5i)T + (5.00e12 + 3.41e12i)T^{2} \)
71 \( 1 + (-2.91e6 + 2.70e6i)T + (6.79e11 - 9.06e12i)T^{2} \)
73 \( 1 + (-3.82e5 - 5.09e6i)T + (-1.09e13 + 1.64e12i)T^{2} \)
79 \( 1 + (-3.11e6 - 5.39e6i)T + (-9.60e12 + 1.66e13i)T^{2} \)
83 \( 1 + (-3.09e6 + 4.66e5i)T + (2.59e13 - 7.99e12i)T^{2} \)
89 \( 1 + (-7.90e6 + 1.19e6i)T + (4.22e13 - 1.30e13i)T^{2} \)
97 \( 1 + (-7.14e5 + 3.13e6i)T + (-7.27e13 - 3.50e13i)T^{2} \)
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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

−14.64626157719846972219041959913, −13.32465362771597893349522086839, −12.15764619263022789600251983600, −11.37537278142219252215022901793, −10.26371191078867452863250470902, −8.233278174872508766262936519904, −7.20655755124205288938468168512, −4.84678809061374377601027767394, −3.44695900938997645070379555806, −2.29749875056871492916631670986, 0.56383850977255704320575043263, 3.50004641049315000872167655407, 4.90395787918088165295155481218, 6.24860805402447832825026187242, 7.911725381765378164705217651819, 8.727898636165695719632528400256, 10.80019517869883418669711421435, 12.02445093224482448964088929066, 13.43679683664100668577846745889, 14.29878804039497891525091415718

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