Properties

Degree 2
Conductor 43
Sign $0.247 + 0.968i$
Motivic weight 7
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.23 − 1.55i)2-s + (45.7 + 6.89i)3-s + (27.6 − 120. i)4-s + (82.5 + 56.2i)5-s + (−45.9 − 79.5i)6-s + (577. − 1.00e3i)7-s + (−451. + 217. i)8-s + (−47.1 − 14.5i)9-s + (−14.8 − 197. i)10-s + (−55.8 − 244. i)11-s + (2.09e3 − 5.33e3i)12-s + (−226. + 3.02e3i)13-s + (−2.27e3 + 342. i)14-s + (3.38e3 + 3.14e3i)15-s + (−1.34e4 − 6.45e3i)16-s + (7.97e3 − 5.43e3i)17-s + ⋯
L(s)  = 1  + (−0.109 − 0.137i)2-s + (0.977 + 0.147i)3-s + (0.215 − 0.944i)4-s + (0.295 + 0.201i)5-s + (−0.0868 − 0.150i)6-s + (0.636 − 1.10i)7-s + (−0.311 + 0.150i)8-s + (−0.0215 − 0.00665i)9-s + (−0.00469 − 0.0626i)10-s + (−0.0126 − 0.0554i)11-s + (0.350 − 0.891i)12-s + (−0.0285 + 0.381i)13-s + (−0.221 + 0.0333i)14-s + (0.258 + 0.240i)15-s + (−0.818 − 0.394i)16-s + (0.393 − 0.268i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.247 + 0.968i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.247 + 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $0.247 + 0.968i$
motivic weight  =  \(7\)
character  :  $\chi_{43} (9, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 43,\ (\ :7/2),\ 0.247 + 0.968i)\)
\(L(4)\)  \(\approx\)  \(2.00754 - 1.55840i\)
\(L(\frac12)\)  \(\approx\)  \(2.00754 - 1.55840i\)
\(L(\frac{9}{2})\)   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 \( 1 + (4.94e5 - 1.63e5i)T \)
good2 \( 1 + (1.23 + 1.55i)T + (-28.4 + 124. i)T^{2} \)
3 \( 1 + (-45.7 - 6.89i)T + (2.08e3 + 644. i)T^{2} \)
5 \( 1 + (-82.5 - 56.2i)T + (2.85e4 + 7.27e4i)T^{2} \)
7 \( 1 + (-577. + 1.00e3i)T + (-4.11e5 - 7.13e5i)T^{2} \)
11 \( 1 + (55.8 + 244. i)T + (-1.75e7 + 8.45e6i)T^{2} \)
13 \( 1 + (226. - 3.02e3i)T + (-6.20e7 - 9.35e6i)T^{2} \)
17 \( 1 + (-7.97e3 + 5.43e3i)T + (1.49e8 - 3.81e8i)T^{2} \)
19 \( 1 + (-7.82e3 + 2.41e3i)T + (7.38e8 - 5.03e8i)T^{2} \)
23 \( 1 + (-5.80e4 + 5.38e4i)T + (2.54e8 - 3.39e9i)T^{2} \)
29 \( 1 + (-9.63e4 + 1.45e4i)T + (1.64e10 - 5.08e9i)T^{2} \)
31 \( 1 + (-2.95e3 + 7.52e3i)T + (-2.01e10 - 1.87e10i)T^{2} \)
37 \( 1 + (-2.04e5 - 3.54e5i)T + (-4.74e10 + 8.22e10i)T^{2} \)
41 \( 1 + (-4.21e5 - 5.28e5i)T + (-4.33e10 + 1.89e11i)T^{2} \)
47 \( 1 + (1.74e5 - 7.64e5i)T + (-4.56e11 - 2.19e11i)T^{2} \)
53 \( 1 + (1.42e4 + 1.90e5i)T + (-1.16e12 + 1.75e11i)T^{2} \)
59 \( 1 + (9.34e5 + 4.50e5i)T + (1.55e12 + 1.94e12i)T^{2} \)
61 \( 1 + (-8.49e5 - 2.16e6i)T + (-2.30e12 + 2.13e12i)T^{2} \)
67 \( 1 + (-2.41e5 + 7.45e4i)T + (5.00e12 - 3.41e12i)T^{2} \)
71 \( 1 + (-5.80e5 - 5.38e5i)T + (6.79e11 + 9.06e12i)T^{2} \)
73 \( 1 + (-1.83e5 + 2.44e6i)T + (-1.09e13 - 1.64e12i)T^{2} \)
79 \( 1 + (1.19e6 - 2.06e6i)T + (-9.60e12 - 1.66e13i)T^{2} \)
83 \( 1 + (-3.42e6 - 5.16e5i)T + (2.59e13 + 7.99e12i)T^{2} \)
89 \( 1 + (-7.37e6 - 1.11e6i)T + (4.22e13 + 1.30e13i)T^{2} \)
97 \( 1 + (-2.92e6 - 1.28e7i)T + (-7.27e13 + 3.50e13i)T^{2} \)
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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

−14.31334436677547626753703366788, −13.54694624026303137336412585355, −11.50787496796206742520457958161, −10.40861937608663372345099832278, −9.407134269773353329501346580363, −8.006501527534795991670954881442, −6.47760560701366901096583622095, −4.62809551039430375796743857970, −2.70613074430272818932315713196, −1.06036995601334335826923411380, 2.08930075018093272185734784756, 3.32567821746712188675370538415, 5.48085957914207766583672551630, 7.47740482032270265756803758911, 8.458461283356907994046969026426, 9.264387699057613890125192649581, 11.32849756922199561663161549649, 12.45334677633498280644225045113, 13.51174946175332754887723025801, 14.76749478395199141718204845449

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