Properties

Degree 2
Conductor 43
Sign $0.552 - 0.833i$
Motivic weight 6
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−6.27 + 13.0i)2-s + (36.1 + 2.71i)3-s + (−90.4 − 113. i)4-s + (22.1 − 71.8i)5-s + (−262. + 454. i)6-s + (414. − 239. i)7-s + (1.14e3 − 260. i)8-s + (580. + 87.5i)9-s + (796. + 739. i)10-s + (846. − 1.06e3i)11-s + (−2.96e3 − 4.34e3i)12-s + (1.57e3 − 1.45e3i)13-s + (517. + 6.90e3i)14-s + (996. − 2.53e3i)15-s + (−1.70e3 + 7.45e3i)16-s + (−5.32e3 + 1.64e3i)17-s + ⋯
L(s)  = 1  + (−0.784 + 1.62i)2-s + (1.34 + 0.100i)3-s + (−1.41 − 1.77i)4-s + (0.177 − 0.574i)5-s + (−1.21 + 2.10i)6-s + (1.20 − 0.698i)7-s + (2.22 − 0.508i)8-s + (0.796 + 0.120i)9-s + (0.796 + 0.739i)10-s + (0.636 − 0.797i)11-s + (−1.71 − 2.51i)12-s + (0.715 − 0.663i)13-s + (0.188 + 2.51i)14-s + (0.295 − 0.752i)15-s + (−0.415 + 1.82i)16-s + (−1.08 + 0.334i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.552 - 0.833i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.552 - 0.833i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $0.552 - 0.833i$
motivic weight  =  \(6\)
character  :  $\chi_{43} (3, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 43,\ (\ :3),\ 0.552 - 0.833i)\)
\(L(\frac{7}{2})\)  \(\approx\)  \(1.67060 + 0.896473i\)
\(L(\frac12)\)  \(\approx\)  \(1.67060 + 0.896473i\)
\(L(4)\)   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 \( 1 + (-7.54e4 - 2.49e4i)T \)
good2 \( 1 + (6.27 - 13.0i)T + (-39.9 - 50.0i)T^{2} \)
3 \( 1 + (-36.1 - 2.71i)T + (720. + 108. i)T^{2} \)
5 \( 1 + (-22.1 + 71.8i)T + (-1.29e4 - 8.80e3i)T^{2} \)
7 \( 1 + (-414. + 239. i)T + (5.88e4 - 1.01e5i)T^{2} \)
11 \( 1 + (-846. + 1.06e3i)T + (-3.94e5 - 1.72e6i)T^{2} \)
13 \( 1 + (-1.57e3 + 1.45e3i)T + (3.60e5 - 4.81e6i)T^{2} \)
17 \( 1 + (5.32e3 - 1.64e3i)T + (1.99e7 - 1.35e7i)T^{2} \)
19 \( 1 + (-1.49e3 - 9.94e3i)T + (-4.49e7 + 1.38e7i)T^{2} \)
23 \( 1 + (6.76e3 + 1.72e4i)T + (-1.08e8 + 1.00e8i)T^{2} \)
29 \( 1 + (1.82e4 - 1.36e3i)T + (5.88e8 - 8.86e7i)T^{2} \)
31 \( 1 + (3.34e3 - 2.28e3i)T + (3.24e8 - 8.26e8i)T^{2} \)
37 \( 1 + (-4.27e4 - 2.46e4i)T + (1.28e9 + 2.22e9i)T^{2} \)
41 \( 1 + (-5.21e4 - 2.51e4i)T + (2.96e9 + 3.71e9i)T^{2} \)
47 \( 1 + (-6.94e4 - 8.70e4i)T + (-2.39e9 + 1.05e10i)T^{2} \)
53 \( 1 + (1.00e5 + 9.34e4i)T + (1.65e9 + 2.21e10i)T^{2} \)
59 \( 1 + (5.58e4 - 2.44e5i)T + (-3.80e10 - 1.83e10i)T^{2} \)
61 \( 1 + (-1.68e5 + 2.47e5i)T + (-1.88e10 - 4.79e10i)T^{2} \)
67 \( 1 + (5.41e5 - 8.15e4i)T + (8.64e10 - 2.66e10i)T^{2} \)
71 \( 1 + (3.42e5 + 1.34e5i)T + (9.39e10 + 8.71e10i)T^{2} \)
73 \( 1 + (-3.07e5 - 3.31e5i)T + (-1.13e10 + 1.50e11i)T^{2} \)
79 \( 1 + (1.87e5 + 3.25e5i)T + (-1.21e11 + 2.10e11i)T^{2} \)
83 \( 1 + (1.72e4 - 2.29e5i)T + (-3.23e11 - 4.87e10i)T^{2} \)
89 \( 1 + (-8.99e5 - 6.73e4i)T + (4.91e11 + 7.40e10i)T^{2} \)
97 \( 1 + (1.51e5 - 1.90e5i)T + (-1.85e11 - 8.12e11i)T^{2} \)
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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

−14.70667281597611723110120037810, −14.34606496933555004238333308127, −13.30874152524467510023993321018, −10.72862916888103756319634409392, −9.184973987316672554294419712926, −8.382334835445196689010704086367, −7.79862082327268011770042389687, −6.02258665242521236968548437519, −4.26564066665255418101302809957, −1.19853047833942677479436659361, 1.75604384346521729375396161968, 2.55540872917980956502571841808, 4.16166727282693691811877265832, 7.52387914392897467036607469423, 8.930462775491159292990113259048, 9.263408512549201363101731738115, 11.01288926038637676172756169871, 11.72457414906974547672431265103, 13.27340484176564590566160410772, 14.20444315216675691944031258002

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