Properties

Degree 2
Conductor 43
Sign $0.374 + 0.927i$
Motivic weight 7
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

\( d \)  =  \(2\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $0.374 + 0.927i$
motivic weight  =  \(7\)
character  :  $\chi_{43} (9, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 43,\ (\ :7/2),\ 0.374 + 0.927i)\)
\(L(4)\)  \(\approx\)  \(1.46364 - 0.987635i\)
\(L(\frac12)\)  \(\approx\)  \(1.46364 - 0.987635i\)
\(L(\frac{9}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \neq 43$,\(F_p(T)\) is a polynomial of degree 2. If $p = 43$, then $F_p(T)$ is a polynomial of degree at most 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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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

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

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