Properties

Degree 2
Conductor 43
Sign $-0.320 + 0.947i$
Motivic weight 5
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−5.37 + 6.73i)2-s + (15.6 + 19.5i)3-s + (−9.39 − 41.1i)4-s + (−42.9 − 20.6i)5-s − 215.·6-s − 55.8·7-s + (79.3 + 38.1i)8-s + (−85.6 + 375. i)9-s + (369. − 177. i)10-s + (−41.5 + 182. i)11-s + (659. − 827. i)12-s + (−716. − 344. i)13-s + (299. − 376. i)14-s + (−265. − 1.16e3i)15-s + (533. − 257. i)16-s + (820. − 394. i)17-s + ⋯
L(s)  = 1  + (−0.949 + 1.19i)2-s + (1.00 + 1.25i)3-s + (−0.293 − 1.28i)4-s + (−0.767 − 0.369i)5-s − 2.44·6-s − 0.430·7-s + (0.438 + 0.211i)8-s + (−0.352 + 1.54i)9-s + (1.16 − 0.562i)10-s + (−0.103 + 0.454i)11-s + (1.32 − 1.65i)12-s + (−1.17 − 0.566i)13-s + (0.409 − 0.512i)14-s + (−0.304 − 1.33i)15-s + (0.521 − 0.251i)16-s + (0.688 − 0.331i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.320 + 0.947i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.320 + 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $-0.320 + 0.947i$
motivic weight  =  \(5\)
character  :  $\chi_{43} (4, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 43,\ (\ :5/2),\ -0.320 + 0.947i)\)
\(L(3)\)  \(\approx\)  \(0.312846 - 0.435974i\)
\(L(\frac12)\)  \(\approx\)  \(0.312846 - 0.435974i\)
\(L(\frac{7}{2})\)   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 \( 1 + (-5.57e3 + 1.07e4i)T \)
good2 \( 1 + (5.37 - 6.73i)T + (-7.12 - 31.1i)T^{2} \)
3 \( 1 + (-15.6 - 19.5i)T + (-54.0 + 236. i)T^{2} \)
5 \( 1 + (42.9 + 20.6i)T + (1.94e3 + 2.44e3i)T^{2} \)
7 \( 1 + 55.8T + 1.68e4T^{2} \)
11 \( 1 + (41.5 - 182. i)T + (-1.45e5 - 6.98e4i)T^{2} \)
13 \( 1 + (716. + 344. i)T + (2.31e5 + 2.90e5i)T^{2} \)
17 \( 1 + (-820. + 394. i)T + (8.85e5 - 1.11e6i)T^{2} \)
19 \( 1 + (-255. - 1.11e3i)T + (-2.23e6 + 1.07e6i)T^{2} \)
23 \( 1 + (1.08e3 - 4.77e3i)T + (-5.79e6 - 2.79e6i)T^{2} \)
29 \( 1 + (623. - 782. i)T + (-4.56e6 - 1.99e7i)T^{2} \)
31 \( 1 + (-854. + 1.07e3i)T + (-6.37e6 - 2.79e7i)T^{2} \)
37 \( 1 + 1.41e4T + 6.93e7T^{2} \)
41 \( 1 + (1.02e4 - 1.28e4i)T + (-2.57e7 - 1.12e8i)T^{2} \)
47 \( 1 + (-4.58e3 - 2.01e4i)T + (-2.06e8 + 9.95e7i)T^{2} \)
53 \( 1 + (-1.81e4 + 8.71e3i)T + (2.60e8 - 3.26e8i)T^{2} \)
59 \( 1 + (-4.50e3 + 2.17e3i)T + (4.45e8 - 5.58e8i)T^{2} \)
61 \( 1 + (2.17e4 + 2.73e4i)T + (-1.87e8 + 8.23e8i)T^{2} \)
67 \( 1 + (-1.51e4 - 6.65e4i)T + (-1.21e9 + 5.85e8i)T^{2} \)
71 \( 1 + (3.10e3 + 1.35e4i)T + (-1.62e9 + 7.82e8i)T^{2} \)
73 \( 1 + (-4.58e4 - 2.20e4i)T + (1.29e9 + 1.62e9i)T^{2} \)
79 \( 1 - 4.47e4T + 3.07e9T^{2} \)
83 \( 1 + (-2.76e4 - 3.47e4i)T + (-8.76e8 + 3.84e9i)T^{2} \)
89 \( 1 + (2.01e4 + 2.52e4i)T + (-1.24e9 + 5.44e9i)T^{2} \)
97 \( 1 + (2.10e4 - 9.22e4i)T + (-7.73e9 - 3.72e9i)T^{2} \)
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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

−15.69778512418818718119532465416, −15.18374948883173564197377050935, −14.14018392123098067111564795526, −12.19413533991042520876682873312, −10.02870620550899682286238644409, −9.552412279673045477559678192762, −8.240997667632548733020585987466, −7.46117604809850996304589457742, −5.22902756885025302207501475347, −3.48729370649887662904176238615, 0.33855231723819568864670057986, 2.16517192286703372078652140492, 3.32161228023824499488928080048, 6.93017970776983003584402555942, 8.055473311616410486268656973534, 9.034626105609373178109254456071, 10.42594537019777529867686110554, 11.92435728195949557980360272417, 12.49959500503104910528691955625, 13.90212851607453621094942701904

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