Properties

Degree 2
Conductor 43
Sign $0.865 + 0.500i$
Motivic weight 4
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.410 + 0.852i)2-s + (−16.8 − 1.26i)3-s + (9.41 + 11.8i)4-s + (6.39 − 20.7i)5-s + (7.98 − 13.8i)6-s + (31.1 − 17.9i)7-s + (−28.7 + 6.55i)8-s + (201. + 30.3i)9-s + (15.0 + 13.9i)10-s + (139. − 174. i)11-s + (−143. − 210. i)12-s + (−60.7 + 56.4i)13-s + (2.54 + 33.9i)14-s + (−133. + 340. i)15-s + (−47.5 + 208. i)16-s + (67.3 − 20.7i)17-s + ⋯
L(s)  = 1  + (−0.102 + 0.213i)2-s + (−1.86 − 0.140i)3-s + (0.588 + 0.738i)4-s + (0.255 − 0.828i)5-s + (0.221 − 0.383i)6-s + (0.636 − 0.367i)7-s + (−0.448 + 0.102i)8-s + (2.48 + 0.374i)9-s + (0.150 + 0.139i)10-s + (1.15 − 1.44i)11-s + (−0.996 − 1.46i)12-s + (−0.359 + 0.333i)13-s + (0.0129 + 0.173i)14-s + (−0.593 + 1.51i)15-s + (−0.185 + 0.814i)16-s + (0.232 − 0.0718i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.865 + 0.500i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.865 + 0.500i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $0.865 + 0.500i$
motivic weight  =  \(4\)
character  :  $\chi_{43} (3, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 43,\ (\ :2),\ 0.865 + 0.500i)\)
\(L(\frac{5}{2})\)  \(\approx\)  \(0.954434 - 0.256328i\)
\(L(\frac12)\)  \(\approx\)  \(0.954434 - 0.256328i\)
\(L(3)\)   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 \( 1 + (-472. + 1.78e3i)T \)
good2 \( 1 + (0.410 - 0.852i)T + (-9.97 - 12.5i)T^{2} \)
3 \( 1 + (16.8 + 1.26i)T + (80.0 + 12.0i)T^{2} \)
5 \( 1 + (-6.39 + 20.7i)T + (-516. - 352. i)T^{2} \)
7 \( 1 + (-31.1 + 17.9i)T + (1.20e3 - 2.07e3i)T^{2} \)
11 \( 1 + (-139. + 174. i)T + (-3.25e3 - 1.42e4i)T^{2} \)
13 \( 1 + (60.7 - 56.4i)T + (2.13e3 - 2.84e4i)T^{2} \)
17 \( 1 + (-67.3 + 20.7i)T + (6.90e4 - 4.70e4i)T^{2} \)
19 \( 1 + (62.0 + 411. i)T + (-1.24e5 + 3.84e4i)T^{2} \)
23 \( 1 + (57.4 + 146. i)T + (-2.05e5 + 1.90e5i)T^{2} \)
29 \( 1 + (-1.20e3 + 90.5i)T + (6.99e5 - 1.05e5i)T^{2} \)
31 \( 1 + (-571. + 389. i)T + (3.37e5 - 8.59e5i)T^{2} \)
37 \( 1 + (664. + 383. i)T + (9.37e5 + 1.62e6i)T^{2} \)
41 \( 1 + (1.31e3 + 633. i)T + (1.76e6 + 2.20e6i)T^{2} \)
47 \( 1 + (-1.22e3 - 1.54e3i)T + (-1.08e6 + 4.75e6i)T^{2} \)
53 \( 1 + (-2.80e3 - 2.60e3i)T + (5.89e5 + 7.86e6i)T^{2} \)
59 \( 1 + (-269. + 1.18e3i)T + (-1.09e7 - 5.25e6i)T^{2} \)
61 \( 1 + (1.13e3 - 1.66e3i)T + (-5.05e6 - 1.28e7i)T^{2} \)
67 \( 1 + (7.96e3 - 1.20e3i)T + (1.92e7 - 5.93e6i)T^{2} \)
71 \( 1 + (3.94e3 + 1.54e3i)T + (1.86e7 + 1.72e7i)T^{2} \)
73 \( 1 + (-876. - 944. i)T + (-2.12e6 + 2.83e7i)T^{2} \)
79 \( 1 + (-4.21e3 - 7.30e3i)T + (-1.94e7 + 3.37e7i)T^{2} \)
83 \( 1 + (-121. + 1.62e3i)T + (-4.69e7 - 7.07e6i)T^{2} \)
89 \( 1 + (4.17e3 + 312. i)T + (6.20e7 + 9.35e6i)T^{2} \)
97 \( 1 + (-577. + 724. i)T + (-1.96e7 - 8.63e7i)T^{2} \)
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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

−15.71593636824541203538504687301, −13.66467041354691335426070051577, −12.29360060922052833774823962733, −11.65339048522799485751866279772, −10.75177525772356319032981814577, −8.765679477781908858823382958614, −7.06244778226089085558783908598, −5.98516237470700873459823665086, −4.51851914767435032506758020658, −0.954144164416155490025254721793, 1.54557215560893486417476078469, 4.85178816890912048617625373686, 6.17159580006454174686629838039, 6.99401450466819765799592018123, 9.985437009913409787517882782868, 10.45219271340441393986506346961, 11.79807004560532194904199218179, 12.15121472832268333112071154996, 14.54622005739224714701204505712, 15.28059595382354072558144697116

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