Properties

Degree 2
Conductor 43
Sign $-0.259 + 0.965i$
Motivic weight 9
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−16.1 − 20.2i)2-s + (−257. − 38.7i)3-s + (−35.1 + 153. i)4-s + (−1.37e3 − 934. i)5-s + (3.36e3 + 5.83e3i)6-s + (71.3 − 123. i)7-s + (−8.25e3 + 3.97e3i)8-s + (4.59e4 + 1.41e4i)9-s + (3.20e3 + 4.27e4i)10-s + (5.12e3 + 2.24e4i)11-s + (1.50e4 − 3.82e4i)12-s + (−614. + 8.19e3i)13-s + (−3.65e3 + 550. i)14-s + (3.16e5 + 2.93e5i)15-s + (2.86e5 + 1.37e5i)16-s + (−1.49e5 + 1.01e5i)17-s + ⋯
L(s)  = 1  + (−0.713 − 0.894i)2-s + (−1.83 − 0.276i)3-s + (−0.0685 + 0.300i)4-s + (−0.980 − 0.668i)5-s + (1.06 + 1.83i)6-s + (0.0112 − 0.0194i)7-s + (−0.712 + 0.343i)8-s + (2.33 + 0.720i)9-s + (0.101 + 1.35i)10-s + (0.105 + 0.462i)11-s + (0.208 − 0.532i)12-s + (−0.00596 + 0.0796i)13-s + (−0.0254 + 0.00383i)14-s + (1.61 + 1.49i)15-s + (1.09 + 0.526i)16-s + (−0.434 + 0.295i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.259 + 0.965i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.259 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $-0.259 + 0.965i$
motivic weight  =  \(9\)
character  :  $\chi_{43} (9, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 43,\ (\ :9/2),\ -0.259 + 0.965i)\)
\(L(5)\)  \(\approx\)  \(0.152821 - 0.199395i\)
\(L(\frac12)\)  \(\approx\)  \(0.152821 - 0.199395i\)
\(L(\frac{11}{2})\)   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 \( 1 + (-2.19e7 - 4.32e6i)T \)
good2 \( 1 + (16.1 + 20.2i)T + (-113. + 499. i)T^{2} \)
3 \( 1 + (257. + 38.7i)T + (1.88e4 + 5.80e3i)T^{2} \)
5 \( 1 + (1.37e3 + 934. i)T + (7.13e5 + 1.81e6i)T^{2} \)
7 \( 1 + (-71.3 + 123. i)T + (-2.01e7 - 3.49e7i)T^{2} \)
11 \( 1 + (-5.12e3 - 2.24e4i)T + (-2.12e9 + 1.02e9i)T^{2} \)
13 \( 1 + (614. - 8.19e3i)T + (-1.04e10 - 1.58e9i)T^{2} \)
17 \( 1 + (1.49e5 - 1.01e5i)T + (4.33e10 - 1.10e11i)T^{2} \)
19 \( 1 + (3.19e5 - 9.84e4i)T + (2.66e11 - 1.81e11i)T^{2} \)
23 \( 1 + (-8.47e5 + 7.86e5i)T + (1.34e11 - 1.79e12i)T^{2} \)
29 \( 1 + (6.57e6 - 9.91e5i)T + (1.38e13 - 4.27e12i)T^{2} \)
31 \( 1 + (2.85e6 - 7.27e6i)T + (-1.93e13 - 1.79e13i)T^{2} \)
37 \( 1 + (4.64e6 + 8.04e6i)T + (-6.49e13 + 1.12e14i)T^{2} \)
41 \( 1 + (-4.17e6 - 5.23e6i)T + (-7.28e13 + 3.19e14i)T^{2} \)
47 \( 1 + (-1.12e7 + 4.91e7i)T + (-1.00e15 - 4.85e14i)T^{2} \)
53 \( 1 + (-5.28e6 - 7.04e7i)T + (-3.26e15 + 4.91e14i)T^{2} \)
59 \( 1 + (9.92e7 + 4.77e7i)T + (5.40e15 + 6.77e15i)T^{2} \)
61 \( 1 + (-7.22e7 - 1.84e8i)T + (-8.57e15 + 7.95e15i)T^{2} \)
67 \( 1 + (1.28e8 - 3.95e7i)T + (2.24e16 - 1.53e16i)T^{2} \)
71 \( 1 + (-5.11e7 - 4.75e7i)T + (3.42e15 + 4.57e16i)T^{2} \)
73 \( 1 + (-1.33e7 + 1.77e8i)T + (-5.82e16 - 8.77e15i)T^{2} \)
79 \( 1 + (-2.43e8 + 4.21e8i)T + (-5.99e16 - 1.03e17i)T^{2} \)
83 \( 1 + (4.94e8 + 7.45e7i)T + (1.78e17 + 5.51e16i)T^{2} \)
89 \( 1 + (7.10e8 + 1.07e8i)T + (3.34e17 + 1.03e17i)T^{2} \)
97 \( 1 + (3.06e7 + 1.34e8i)T + (-6.84e17 + 3.29e17i)T^{2} \)
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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

−12.67489443529073314240056762858, −12.20577742686077759416449475481, −11.16044925322352662241864066887, −10.51876992122094510959611115363, −8.940517646346057088678711517682, −7.22212366859339690990358160143, −5.69861281292664568716406281623, −4.32530095304933295561496469459, −1.58169121268547774490285760639, −0.39679190416133934799858095148, 0.42711177899029604177052243262, 3.86831101298533017544993413610, 5.62738998054150566175600769252, 6.74016865367188722217641578700, 7.62712502904418222867741866272, 9.432203298176215606011467318853, 10.98743734167548925036124088413, 11.52638828270248232212049683084, 12.78719942886367525144123434439, 15.14193667934371503749090543584

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