Properties

Degree 2
Conductor 43
Sign $-0.879 - 0.475i$
Motivic weight 9
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 17.1·2-s + (−66.7 + 115. i)3-s − 217.·4-s + (−131. + 227. i)5-s + (1.14e3 − 1.98e3i)6-s + (1.32e3 + 2.29e3i)7-s + 1.25e4·8-s + (917. + 1.58e3i)9-s + (2.24e3 − 3.89e3i)10-s + 4.42e4·11-s + (1.45e4 − 2.52e4i)12-s + (7.52e4 + 1.30e5i)13-s + (−2.27e4 − 3.93e4i)14-s + (−1.75e4 − 3.03e4i)15-s − 1.03e5·16-s + (2.93e3 + 5.08e3i)17-s + ⋯
L(s)  = 1  − 0.757·2-s + (−0.476 + 0.824i)3-s − 0.425·4-s + (−0.0937 + 0.162i)5-s + (0.360 − 0.625i)6-s + (0.208 + 0.361i)7-s + 1.08·8-s + (0.0466 + 0.0807i)9-s + (0.0710 − 0.123i)10-s + 0.911·11-s + (0.202 − 0.350i)12-s + (0.731 + 1.26i)13-s + (−0.158 − 0.273i)14-s + (−0.0893 − 0.154i)15-s − 0.393·16-s + (0.00852 + 0.0147i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.879 - 0.475i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.879 - 0.475i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $-0.879 - 0.475i$
motivic weight  =  \(9\)
character  :  $\chi_{43} (6, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 43,\ (\ :9/2),\ -0.879 - 0.475i)\)
\(L(5)\)  \(\approx\)  \(0.206255 + 0.814559i\)
\(L(\frac12)\)  \(\approx\)  \(0.206255 + 0.814559i\)
\(L(\frac{11}{2})\)   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 \( 1 + (-2.17e7 + 5.46e6i)T \)
good2 \( 1 + 17.1T + 512T^{2} \)
3 \( 1 + (66.7 - 115. i)T + (-9.84e3 - 1.70e4i)T^{2} \)
5 \( 1 + (131. - 227. i)T + (-9.76e5 - 1.69e6i)T^{2} \)
7 \( 1 + (-1.32e3 - 2.29e3i)T + (-2.01e7 + 3.49e7i)T^{2} \)
11 \( 1 - 4.42e4T + 2.35e9T^{2} \)
13 \( 1 + (-7.52e4 - 1.30e5i)T + (-5.30e9 + 9.18e9i)T^{2} \)
17 \( 1 + (-2.93e3 - 5.08e3i)T + (-5.92e10 + 1.02e11i)T^{2} \)
19 \( 1 + (-2.62e5 + 4.55e5i)T + (-1.61e11 - 2.79e11i)T^{2} \)
23 \( 1 + (-3.89e4 + 6.73e4i)T + (-9.00e11 - 1.55e12i)T^{2} \)
29 \( 1 + (-4.48e5 - 7.77e5i)T + (-7.25e12 + 1.25e13i)T^{2} \)
31 \( 1 + (2.80e6 - 4.85e6i)T + (-1.32e13 - 2.28e13i)T^{2} \)
37 \( 1 + (5.36e6 - 9.29e6i)T + (-6.49e13 - 1.12e14i)T^{2} \)
41 \( 1 - 4.03e6T + 3.27e14T^{2} \)
47 \( 1 + 3.12e7T + 1.11e15T^{2} \)
53 \( 1 + (6.39e6 - 1.10e7i)T + (-1.64e15 - 2.85e15i)T^{2} \)
59 \( 1 - 1.71e8T + 8.66e15T^{2} \)
61 \( 1 + (1.63e7 + 2.82e7i)T + (-5.84e15 + 1.01e16i)T^{2} \)
67 \( 1 + (9.02e7 - 1.56e8i)T + (-1.36e16 - 2.35e16i)T^{2} \)
71 \( 1 + (6.61e6 + 1.14e7i)T + (-2.29e16 + 3.97e16i)T^{2} \)
73 \( 1 + (1.15e8 + 2.00e8i)T + (-2.94e16 + 5.09e16i)T^{2} \)
79 \( 1 + (7.11e7 + 1.23e8i)T + (-5.99e16 + 1.03e17i)T^{2} \)
83 \( 1 + (1.21e8 - 2.11e8i)T + (-9.34e16 - 1.61e17i)T^{2} \)
89 \( 1 + (3.13e8 - 5.43e8i)T + (-1.75e17 - 3.03e17i)T^{2} \)
97 \( 1 + 7.37e8T + 7.60e17T^{2} \)
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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

−14.55466661874557808768134799837, −13.42134168402458664092341720592, −11.63969053094562583481737903754, −10.74519724860429090012572798092, −9.470902148666821153275169732226, −8.719055960640165965237126304233, −6.98128691484620953947039133036, −5.11660849125113149158127537245, −3.96033025512562569508088850921, −1.40776608166575523712075281612, 0.51092867654867049081591885383, 1.32989257540581793955915767765, 3.97980032921206248828434808904, 5.82769164793445089417673369783, 7.33351809045652632020677507885, 8.380677813311504576290847735619, 9.721737022763391701319223404143, 10.98591562169912607400360200692, 12.34894403614280991497515819418, 13.29915803246847498439338489178

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