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

\( d \)  =  \(2\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $-0.933 + 0.357i$
motivic weight  =  \(3\)
character  :  $\chi_{43} (9, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 43,\ (\ :3/2),\ -0.933 + 0.357i)\)
\(L(2)\)  \(\approx\)  \(0.127686 - 0.689694i\)
\(L(\frac12)\)  \(\approx\)  \(0.127686 - 0.689694i\)
\(L(\frac{5}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]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 \( 1 + (118. + 255. i)T \)
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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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

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

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