Properties

Degree $2$
Conductor $43$
Sign $-0.933 - 0.357i$
Motivic weight $3$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.84 + 2.31i)2-s + (0.851 − 0.128i)3-s + (−0.170 − 0.748i)4-s + (−8.91 + 6.07i)5-s + (−1.27 + 2.20i)6-s + (4.30 + 7.45i)7-s + (−19.2 − 9.29i)8-s + (−25.0 + 7.73i)9-s + (2.38 − 31.8i)10-s + (−0.134 + 0.587i)11-s + (−0.241 − 0.615i)12-s + (2.09 + 27.9i)13-s + (−25.1 − 3.79i)14-s + (−6.81 + 6.32i)15-s + (62.6 − 30.1i)16-s + (55.0 + 37.5i)17-s + ⋯
L(s)  = 1  + (−0.652 + 0.818i)2-s + (0.163 − 0.0247i)3-s + (−0.0213 − 0.0935i)4-s + (−0.796 + 0.543i)5-s + (−0.0867 + 0.150i)6-s + (0.232 + 0.402i)7-s + (−0.852 − 0.410i)8-s + (−0.929 + 0.286i)9-s + (0.0754 − 1.00i)10-s + (−0.00367 + 0.0160i)11-s + (−0.00581 − 0.0148i)12-s + (0.0446 + 0.595i)13-s + (−0.480 − 0.0724i)14-s + (−0.117 + 0.108i)15-s + (0.979 − 0.471i)16-s + (0.785 + 0.535i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.933 - 0.357i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.933 - 0.357i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(43\)
Sign: $-0.933 - 0.357i$
Motivic weight: \(3\)
Character: $\chi_{43} (24, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 43,\ (\ :3/2),\ -0.933 - 0.357i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.127686 + 0.689694i\)
\(L(\frac12)\) \(\approx\) \(0.127686 + 0.689694i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad43 \( 1 + (118. - 255. i)T \)
good2 \( 1 + (1.84 - 2.31i)T + (-1.78 - 7.79i)T^{2} \)
3 \( 1 + (-0.851 + 0.128i)T + (25.8 - 7.95i)T^{2} \)
5 \( 1 + (8.91 - 6.07i)T + (45.6 - 116. i)T^{2} \)
7 \( 1 + (-4.30 - 7.45i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + (0.134 - 0.587i)T + (-1.19e3 - 577. i)T^{2} \)
13 \( 1 + (-2.09 - 27.9i)T + (-2.17e3 + 327. i)T^{2} \)
17 \( 1 + (-55.0 - 37.5i)T + (1.79e3 + 4.57e3i)T^{2} \)
19 \( 1 + (-154. - 47.6i)T + (5.66e3 + 3.86e3i)T^{2} \)
23 \( 1 + (33.1 + 30.7i)T + (909. + 1.21e4i)T^{2} \)
29 \( 1 + (46.8 + 7.06i)T + (2.33e4 + 7.18e3i)T^{2} \)
31 \( 1 + (4.41 + 11.2i)T + (-2.18e4 + 2.02e4i)T^{2} \)
37 \( 1 + (-94.0 + 162. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (40.0 - 50.2i)T + (-1.53e4 - 6.71e4i)T^{2} \)
47 \( 1 + (-46.8 - 205. i)T + (-9.35e4 + 4.50e4i)T^{2} \)
53 \( 1 + (33.4 - 446. i)T + (-1.47e5 - 2.21e4i)T^{2} \)
59 \( 1 + (151. - 72.7i)T + (1.28e5 - 1.60e5i)T^{2} \)
61 \( 1 + (-282. + 720. i)T + (-1.66e5 - 1.54e5i)T^{2} \)
67 \( 1 + (787. + 243. i)T + (2.48e5 + 1.69e5i)T^{2} \)
71 \( 1 + (-298. + 277. i)T + (2.67e4 - 3.56e5i)T^{2} \)
73 \( 1 + (-39.4 - 525. i)T + (-3.84e5 + 5.79e4i)T^{2} \)
79 \( 1 + (-363. - 629. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (639. - 96.4i)T + (5.46e5 - 1.68e5i)T^{2} \)
89 \( 1 + (-314. + 47.3i)T + (6.73e5 - 2.07e5i)T^{2} \)
97 \( 1 + (52.1 - 228. i)T + (-8.22e5 - 3.95e5i)T^{2} \)
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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

−16.08700511908312146667386746998, −15.01568018931840267871910183048, −14.12735149107458803887441422081, −12.19899679674776179191111945952, −11.32080423876488196815728091796, −9.504566377257196804926455065796, −8.203226960737269887374622413866, −7.43626333007262433894357403656, −5.82445311201606165810133942174, −3.31371460907740431154323900371, 0.70095771137284383648961682888, 3.21417236010353470602843127616, 5.42766773325665995833689979388, 7.72028587757717726026534714499, 8.882373700084850871406604259886, 10.06479608852029580170251790403, 11.48864841929333805921051631383, 12.00964721860399089817076637829, 13.77819820679042807651103596681, 14.99220188230406685326630382422

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