Properties

Degree 2
Conductor 43
Sign $0.952 + 0.303i$
Motivic weight 7
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−7.47 − 9.37i)2-s + (−35.4 − 5.33i)3-s + (−3.49 + 15.2i)4-s + (−37.7 − 25.7i)5-s + (214. + 371. i)6-s + (−223. + 387. i)7-s + (−1.21e3 + 584. i)8-s + (−864. − 266. i)9-s + (40.9 + 546. i)10-s + (1.76e3 + 7.74e3i)11-s + (205. − 523. i)12-s + (748. − 9.98e3i)13-s + (5.30e3 − 799. i)14-s + (1.20e3 + 1.11e3i)15-s + (1.63e4 + 7.87e3i)16-s + (1.41e4 − 9.65e3i)17-s + ⋯
L(s)  = 1  + (−0.660 − 0.828i)2-s + (−0.757 − 0.114i)3-s + (−0.0272 + 0.119i)4-s + (−0.135 − 0.0921i)5-s + (0.405 + 0.702i)6-s + (−0.246 + 0.427i)7-s + (−0.837 + 0.403i)8-s + (−0.395 − 0.121i)9-s + (0.0129 + 0.172i)10-s + (0.400 + 1.75i)11-s + (0.0343 − 0.0873i)12-s + (0.0944 − 1.26i)13-s + (0.516 − 0.0778i)14-s + (0.0918 + 0.0852i)15-s + (0.997 + 0.480i)16-s + (0.699 − 0.476i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.952 + 0.303i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.952 + 0.303i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $0.952 + 0.303i$
motivic weight  =  \(7\)
character  :  $\chi_{43} (9, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 43,\ (\ :7/2),\ 0.952 + 0.303i)\)
\(L(4)\)  \(\approx\)  \(0.644904 - 0.100113i\)
\(L(\frac12)\)  \(\approx\)  \(0.644904 - 0.100113i\)
\(L(\frac{9}{2})\)   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 \( 1 + (1.23e5 - 5.06e5i)T \)
good2 \( 1 + (7.47 + 9.37i)T + (-28.4 + 124. i)T^{2} \)
3 \( 1 + (35.4 + 5.33i)T + (2.08e3 + 644. i)T^{2} \)
5 \( 1 + (37.7 + 25.7i)T + (2.85e4 + 7.27e4i)T^{2} \)
7 \( 1 + (223. - 387. i)T + (-4.11e5 - 7.13e5i)T^{2} \)
11 \( 1 + (-1.76e3 - 7.74e3i)T + (-1.75e7 + 8.45e6i)T^{2} \)
13 \( 1 + (-748. + 9.98e3i)T + (-6.20e7 - 9.35e6i)T^{2} \)
17 \( 1 + (-1.41e4 + 9.65e3i)T + (1.49e8 - 3.81e8i)T^{2} \)
19 \( 1 + (1.53e4 - 4.74e3i)T + (7.38e8 - 5.03e8i)T^{2} \)
23 \( 1 + (-4.31e4 + 4.00e4i)T + (2.54e8 - 3.39e9i)T^{2} \)
29 \( 1 + (-1.69e5 + 2.56e4i)T + (1.64e10 - 5.08e9i)T^{2} \)
31 \( 1 + (9.02e4 - 2.30e5i)T + (-2.01e10 - 1.87e10i)T^{2} \)
37 \( 1 + (-2.64e5 - 4.58e5i)T + (-4.74e10 + 8.22e10i)T^{2} \)
41 \( 1 + (4.37e5 + 5.48e5i)T + (-4.33e10 + 1.89e11i)T^{2} \)
47 \( 1 + (-9.85e4 + 4.31e5i)T + (-4.56e11 - 2.19e11i)T^{2} \)
53 \( 1 + (-9.96e4 - 1.32e6i)T + (-1.16e12 + 1.75e11i)T^{2} \)
59 \( 1 + (1.66e4 + 7.99e3i)T + (1.55e12 + 1.94e12i)T^{2} \)
61 \( 1 + (1.38e5 + 3.53e5i)T + (-2.30e12 + 2.13e12i)T^{2} \)
67 \( 1 + (-3.53e6 + 1.08e6i)T + (5.00e12 - 3.41e12i)T^{2} \)
71 \( 1 + (-1.55e6 - 1.44e6i)T + (6.79e11 + 9.06e12i)T^{2} \)
73 \( 1 + (4.17e5 - 5.57e6i)T + (-1.09e13 - 1.64e12i)T^{2} \)
79 \( 1 + (3.83e5 - 6.63e5i)T + (-9.60e12 - 1.66e13i)T^{2} \)
83 \( 1 + (-5.25e6 - 7.91e5i)T + (2.59e13 + 7.99e12i)T^{2} \)
89 \( 1 + (-6.58e6 - 9.92e5i)T + (4.22e13 + 1.30e13i)T^{2} \)
97 \( 1 + (-7.63e5 - 3.34e6i)T + (-7.27e13 + 3.50e13i)T^{2} \)
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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

−14.56939485359679358167292800613, −12.38484183207484653723531567916, −12.12149838236959427253335433514, −10.66697704163871763652652209516, −9.819213700383308574474674027501, −8.443888002787696019259345198984, −6.53907568053308434270559516457, −5.12899790554303065418245935178, −2.72674887200978116132718496393, −0.931841158564404387207563627000, 0.51533382936559822834907663870, 3.55559659089993088077719868409, 5.77136041480408276769867020682, 6.72045644075050857997648095633, 8.196455526390547371257783818519, 9.293445483764881004787480856426, 10.99649658494682400670888631785, 11.80982235265368267036843645997, 13.47742319569032918477038916718, 14.74820774989498557995536640001

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