Properties

Label 2-43-43.9-c7-0-7
Degree $2$
Conductor $43$
Sign $-0.574 + 0.818i$
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  + (−12.0 − 15.1i)2-s + (−73.9 − 11.1i)3-s + (−54.9 + 240. i)4-s + (−26.8 − 18.3i)5-s + (724. + 1.25e3i)6-s + (53.2 − 92.2i)7-s + (2.07e3 − 9.99e2i)8-s + (3.25e3 + 1.00e3i)9-s + (47.0 + 627. i)10-s + (407. + 1.78e3i)11-s + (6.74e3 − 1.71e4i)12-s + (−222. + 2.96e3i)13-s + (−2.03e3 + 307. i)14-s + (1.78e3 + 1.65e3i)15-s + (−1.17e4 − 5.64e3i)16-s + (−2.23e4 + 1.52e4i)17-s + ⋯
L(s)  = 1  + (−1.06 − 1.33i)2-s + (−1.58 − 0.238i)3-s + (−0.429 + 1.88i)4-s + (−0.0960 − 0.0655i)5-s + (1.36 + 2.37i)6-s + (0.0586 − 0.101i)7-s + (1.43 − 0.690i)8-s + (1.48 + 0.458i)9-s + (0.0148 + 0.198i)10-s + (0.0922 + 0.403i)11-s + (1.12 − 2.87i)12-s + (−0.0280 + 0.374i)13-s + (−0.198 + 0.0299i)14-s + (0.136 + 0.126i)15-s + (−0.714 − 0.344i)16-s + (−1.10 + 0.753i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.574 + 0.818i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.574 + 0.818i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(43\)
Sign: $-0.574 + 0.818i$
Analytic conductor: \(13.4325\)
Root analytic conductor: \(3.66504\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{43} (9, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 43,\ (\ :7/2),\ -0.574 + 0.818i)\)

Particular Values

\(L(4)\) \(\approx\) \(0.159879 - 0.307666i\)
\(L(\frac12)\) \(\approx\) \(0.159879 - 0.307666i\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad43 \( 1 + (4.46e5 + 2.69e5i)T \)
good2 \( 1 + (12.0 + 15.1i)T + (-28.4 + 124. i)T^{2} \)
3 \( 1 + (73.9 + 11.1i)T + (2.08e3 + 644. i)T^{2} \)
5 \( 1 + (26.8 + 18.3i)T + (2.85e4 + 7.27e4i)T^{2} \)
7 \( 1 + (-53.2 + 92.2i)T + (-4.11e5 - 7.13e5i)T^{2} \)
11 \( 1 + (-407. - 1.78e3i)T + (-1.75e7 + 8.45e6i)T^{2} \)
13 \( 1 + (222. - 2.96e3i)T + (-6.20e7 - 9.35e6i)T^{2} \)
17 \( 1 + (2.23e4 - 1.52e4i)T + (1.49e8 - 3.81e8i)T^{2} \)
19 \( 1 + (-1.26e4 + 3.91e3i)T + (7.38e8 - 5.03e8i)T^{2} \)
23 \( 1 + (7.97e4 - 7.39e4i)T + (2.54e8 - 3.39e9i)T^{2} \)
29 \( 1 + (-1.19e5 + 1.80e4i)T + (1.64e10 - 5.08e9i)T^{2} \)
31 \( 1 + (-1.07e5 + 2.75e5i)T + (-2.01e10 - 1.87e10i)T^{2} \)
37 \( 1 + (-3.50e4 - 6.07e4i)T + (-4.74e10 + 8.22e10i)T^{2} \)
41 \( 1 + (-2.40e5 - 3.01e5i)T + (-4.33e10 + 1.89e11i)T^{2} \)
47 \( 1 + (-6.18e4 + 2.70e5i)T + (-4.56e11 - 2.19e11i)T^{2} \)
53 \( 1 + (2.66e4 + 3.55e5i)T + (-1.16e12 + 1.75e11i)T^{2} \)
59 \( 1 + (2.05e6 + 9.87e5i)T + (1.55e12 + 1.94e12i)T^{2} \)
61 \( 1 + (-4.93e5 - 1.25e6i)T + (-2.30e12 + 2.13e12i)T^{2} \)
67 \( 1 + (-3.08e6 + 9.51e5i)T + (5.00e12 - 3.41e12i)T^{2} \)
71 \( 1 + (2.64e5 + 2.44e5i)T + (6.79e11 + 9.06e12i)T^{2} \)
73 \( 1 + (-2.43e5 + 3.24e6i)T + (-1.09e13 - 1.64e12i)T^{2} \)
79 \( 1 + (-2.82e6 + 4.89e6i)T + (-9.60e12 - 1.66e13i)T^{2} \)
83 \( 1 + (-3.08e4 - 4.65e3i)T + (2.59e13 + 7.99e12i)T^{2} \)
89 \( 1 + (-3.41e6 - 5.15e5i)T + (4.22e13 + 1.30e13i)T^{2} \)
97 \( 1 + (-2.85e6 - 1.25e7i)T + (-7.27e13 + 3.50e13i)T^{2} \)
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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

−13.36757301324048217237974249829, −12.09532995270674417230070835990, −11.61805311917007690445830964628, −10.58307788941004549104283744647, −9.608581855621282605448253440836, −7.927041880251486839054228054314, −6.25337148277092115389110982656, −4.28597433745929081336737703308, −1.88653086351144700219691281153, −0.44576671657580193612223978712, 0.69947503090270450113760761030, 4.91290270417677432829850747762, 6.05854317939966382120654751350, 6.97383159259101244495144265112, 8.499706146000058934732493879986, 9.933597997614946683207705293196, 10.93068284687821149847742927240, 12.19161364097468279548139175570, 14.09424847370409781799686569355, 15.64693596583564060010042682389

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