Properties

Degree 2
Conductor 43
Sign $-0.662 + 0.749i$
Motivic weight 10
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 46.2i·2-s − 372. i·3-s − 1.11e3·4-s − 2.51e3i·5-s + 1.72e4·6-s − 2.52e4i·7-s − 3.99e3i·8-s − 7.95e4·9-s + 1.16e5·10-s + 7.13e4·11-s + 4.13e5i·12-s + 3.05e5·13-s + 1.16e6·14-s − 9.37e5·15-s − 9.52e5·16-s − 1.86e6·17-s + ⋯
L(s)  = 1  + 1.44i·2-s − 1.53i·3-s − 1.08·4-s − 0.805i·5-s + 2.21·6-s − 1.50i·7-s − 0.121i·8-s − 1.34·9-s + 1.16·10-s + 0.442·11-s + 1.66i·12-s + 0.821·13-s + 2.17·14-s − 1.23·15-s − 0.908·16-s − 1.31·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.662 + 0.749i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (-0.662 + 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $-0.662 + 0.749i$
motivic weight  =  \(10\)
character  :  $\chi_{43} (42, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 43,\ (\ :5),\ -0.662 + 0.749i)\)
\(L(\frac{11}{2})\)  \(\approx\)  \(0.429028 - 0.952176i\)
\(L(\frac12)\)  \(\approx\)  \(0.429028 - 0.952176i\)
\(L(6)\)   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 \( 1 + (-9.73e7 + 1.10e8i)T \)
good2 \( 1 - 46.2iT - 1.02e3T^{2} \)
3 \( 1 + 372. iT - 5.90e4T^{2} \)
5 \( 1 + 2.51e3iT - 9.76e6T^{2} \)
7 \( 1 + 2.52e4iT - 2.82e8T^{2} \)
11 \( 1 - 7.13e4T + 2.59e10T^{2} \)
13 \( 1 - 3.05e5T + 1.37e11T^{2} \)
17 \( 1 + 1.86e6T + 2.01e12T^{2} \)
19 \( 1 - 2.50e5iT - 6.13e12T^{2} \)
23 \( 1 + 7.17e6T + 4.14e13T^{2} \)
29 \( 1 - 7.03e6iT - 4.20e14T^{2} \)
31 \( 1 + 2.56e7T + 8.19e14T^{2} \)
37 \( 1 - 4.75e7iT - 4.80e15T^{2} \)
41 \( 1 - 9.96e7T + 1.34e16T^{2} \)
47 \( 1 - 8.53e7T + 5.25e16T^{2} \)
53 \( 1 + 6.32e8T + 1.74e17T^{2} \)
59 \( 1 + 1.36e9T + 5.11e17T^{2} \)
61 \( 1 - 5.18e8iT - 7.13e17T^{2} \)
67 \( 1 - 7.52e8T + 1.82e18T^{2} \)
71 \( 1 + 2.82e9iT - 3.25e18T^{2} \)
73 \( 1 - 3.11e9iT - 4.29e18T^{2} \)
79 \( 1 - 1.02e9T + 9.46e18T^{2} \)
83 \( 1 + 5.19e9T + 1.55e19T^{2} \)
89 \( 1 + 4.43e9iT - 3.11e19T^{2} \)
97 \( 1 - 6.65e9T + 7.37e19T^{2} \)
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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

−13.60485854472149833859310117373, −12.67089631511967898075055119625, −11.09629591934129727645010664489, −8.879672005715865042050178366221, −7.85574750834568335375179600803, −6.97290555808323524926337955231, −6.10022006018377286948706518482, −4.35231836299352894944248043762, −1.55694021141924796939389574708, −0.33292563962522141714429183426, 2.17292627919230278398397708780, 3.27188211642032757422541923185, 4.39998638049836234868461901273, 6.10362447927653237918164069376, 8.882061514452002969605612828843, 9.548358665726087921880604296646, 10.85759131967838575515878929144, 11.28716760660189927576298665049, 12.58417396877175518934304199648, 14.23615584263011160379709504771

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