Properties

Degree 2
Conductor 43
Sign $-0.752 + 0.658i$
Motivic weight 5
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−6.52 − 8.17i)2-s + (17.3 + 2.61i)3-s + (−17.2 + 75.4i)4-s + (2.46 + 1.68i)5-s + (−91.6 − 158. i)6-s + (94.8 − 164. i)7-s + (427. − 206. i)8-s + (61.2 + 18.9i)9-s + (−2.33 − 31.1i)10-s + (−24.7 − 108. i)11-s + (−495. + 1.26e3i)12-s + (58.7 − 784. i)13-s + (−1.96e3 + 295. i)14-s + (38.3 + 35.5i)15-s + (−2.24e3 − 1.08e3i)16-s + (−707. + 482. i)17-s + ⋯
L(s)  = 1  + (−1.15 − 1.44i)2-s + (1.11 + 0.167i)3-s + (−0.538 + 2.35i)4-s + (0.0441 + 0.0300i)5-s + (−1.03 − 1.80i)6-s + (0.731 − 1.26i)7-s + (2.36 − 1.13i)8-s + (0.252 + 0.0777i)9-s + (−0.00737 − 0.0984i)10-s + (−0.0616 − 0.269i)11-s + (−0.993 + 2.53i)12-s + (0.0964 − 1.28i)13-s + (−2.67 + 0.403i)14-s + (0.0439 + 0.0408i)15-s + (−2.19 − 1.05i)16-s + (−0.593 + 0.404i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.752 + 0.658i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.752 + 0.658i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $-0.752 + 0.658i$
motivic weight  =  \(5\)
character  :  $\chi_{43} (9, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 43,\ (\ :5/2),\ -0.752 + 0.658i)\)
\(L(3)\)  \(\approx\)  \(0.429593 - 1.14362i\)
\(L(\frac12)\)  \(\approx\)  \(0.429593 - 1.14362i\)
\(L(\frac{7}{2})\)   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 \( 1 + (-8.59e3 - 8.55e3i)T \)
good2 \( 1 + (6.52 + 8.17i)T + (-7.12 + 31.1i)T^{2} \)
3 \( 1 + (-17.3 - 2.61i)T + (232. + 71.6i)T^{2} \)
5 \( 1 + (-2.46 - 1.68i)T + (1.14e3 + 2.90e3i)T^{2} \)
7 \( 1 + (-94.8 + 164. i)T + (-8.40e3 - 1.45e4i)T^{2} \)
11 \( 1 + (24.7 + 108. i)T + (-1.45e5 + 6.98e4i)T^{2} \)
13 \( 1 + (-58.7 + 784. i)T + (-3.67e5 - 5.53e4i)T^{2} \)
17 \( 1 + (707. - 482. i)T + (5.18e5 - 1.32e6i)T^{2} \)
19 \( 1 + (-1.76e3 + 545. i)T + (2.04e6 - 1.39e6i)T^{2} \)
23 \( 1 + (1.91e3 - 1.77e3i)T + (4.80e5 - 6.41e6i)T^{2} \)
29 \( 1 + (-7.55e3 + 1.13e3i)T + (1.95e7 - 6.04e6i)T^{2} \)
31 \( 1 + (-1.23e3 + 3.13e3i)T + (-2.09e7 - 1.94e7i)T^{2} \)
37 \( 1 + (-2.71e3 - 4.69e3i)T + (-3.46e7 + 6.00e7i)T^{2} \)
41 \( 1 + (1.08e4 + 1.36e4i)T + (-2.57e7 + 1.12e8i)T^{2} \)
47 \( 1 + (6.31e3 - 2.76e4i)T + (-2.06e8 - 9.95e7i)T^{2} \)
53 \( 1 + (-408. - 5.45e3i)T + (-4.13e8 + 6.23e7i)T^{2} \)
59 \( 1 + (-3.95e4 - 1.90e4i)T + (4.45e8 + 5.58e8i)T^{2} \)
61 \( 1 + (-9.33e3 - 2.37e4i)T + (-6.19e8 + 5.74e8i)T^{2} \)
67 \( 1 + (-3.81e4 + 1.17e4i)T + (1.11e9 - 7.60e8i)T^{2} \)
71 \( 1 + (-1.16e4 - 1.08e4i)T + (1.34e8 + 1.79e9i)T^{2} \)
73 \( 1 + (1.86e3 - 2.49e4i)T + (-2.04e9 - 3.08e8i)T^{2} \)
79 \( 1 + (2.02e3 - 3.49e3i)T + (-1.53e9 - 2.66e9i)T^{2} \)
83 \( 1 + (7.90e3 + 1.19e3i)T + (3.76e9 + 1.16e9i)T^{2} \)
89 \( 1 + (-5.17e4 - 7.79e3i)T + (5.33e9 + 1.64e9i)T^{2} \)
97 \( 1 + (-6.41e3 - 2.80e4i)T + (-7.73e9 + 3.72e9i)T^{2} \)
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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

−14.07046853972112922096502961667, −13.29771217169453023356338737939, −11.72214215211427149899782004317, −10.58527525286408711996529039345, −9.760834354129233764931059817551, −8.339960276134958044537376812190, −7.77861437336504599232788182978, −3.98699303115411448201399415393, −2.70382749939181189845899041456, −0.912941700925557023739668789863, 1.95466564269240668174216321585, 5.20520836312557166825910942862, 6.79640216879306247276191624593, 8.159548432270713044281577941410, 8.787836108317517287627962518803, 9.710489603474449964857055002104, 11.67171393197402527612406279123, 13.87608267560822593605467868020, 14.51450405815819893468380820018, 15.44973280497714038787997085098

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