Properties

Degree 2
Conductor 43
Sign $-0.761 + 0.648i$
Motivic weight 8
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 23.5i·2-s − 46.0i·3-s − 297.·4-s + 373. i·5-s + 1.08e3·6-s + 2.48e3i·7-s − 983. i·8-s + 4.43e3·9-s − 8.79e3·10-s + 7.40e3·11-s + 1.37e4i·12-s − 3.62e4·13-s − 5.85e4·14-s + 1.72e4·15-s − 5.30e4·16-s − 8.88e4·17-s + ⋯
L(s)  = 1  + 1.47i·2-s − 0.568i·3-s − 1.16·4-s + 0.598i·5-s + 0.836·6-s + 1.03i·7-s − 0.240i·8-s + 0.676·9-s − 0.879·10-s + 0.505·11-s + 0.661i·12-s − 1.27·13-s − 1.52·14-s + 0.340·15-s − 0.810·16-s − 1.06·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.761 + 0.648i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.761 + 0.648i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $-0.761 + 0.648i$
motivic weight  =  \(8\)
character  :  $\chi_{43} (42, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 43,\ (\ :4),\ -0.761 + 0.648i)\)
\(L(\frac{9}{2})\)  \(\approx\)  \(0.379940 - 1.03273i\)
\(L(\frac12)\)  \(\approx\)  \(0.379940 - 1.03273i\)
\(L(5)\)   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 \( 1 + (2.60e6 - 2.21e6i)T \)
good2 \( 1 - 23.5iT - 256T^{2} \)
3 \( 1 + 46.0iT - 6.56e3T^{2} \)
5 \( 1 - 373. iT - 3.90e5T^{2} \)
7 \( 1 - 2.48e3iT - 5.76e6T^{2} \)
11 \( 1 - 7.40e3T + 2.14e8T^{2} \)
13 \( 1 + 3.62e4T + 8.15e8T^{2} \)
17 \( 1 + 8.88e4T + 6.97e9T^{2} \)
19 \( 1 + 4.18e4iT - 1.69e10T^{2} \)
23 \( 1 + 2.94e5T + 7.83e10T^{2} \)
29 \( 1 + 4.31e4iT - 5.00e11T^{2} \)
31 \( 1 + 5.57e5T + 8.52e11T^{2} \)
37 \( 1 - 3.70e6iT - 3.51e12T^{2} \)
41 \( 1 - 1.14e6T + 7.98e12T^{2} \)
47 \( 1 + 7.21e5T + 2.38e13T^{2} \)
53 \( 1 + 8.22e6T + 6.22e13T^{2} \)
59 \( 1 - 5.76e6T + 1.46e14T^{2} \)
61 \( 1 + 1.24e7iT - 1.91e14T^{2} \)
67 \( 1 - 2.60e7T + 4.06e14T^{2} \)
71 \( 1 - 3.21e7iT - 6.45e14T^{2} \)
73 \( 1 - 1.56e7iT - 8.06e14T^{2} \)
79 \( 1 - 2.11e7T + 1.51e15T^{2} \)
83 \( 1 + 5.69e7T + 2.25e15T^{2} \)
89 \( 1 - 6.26e7iT - 3.93e15T^{2} \)
97 \( 1 - 1.82e7T + 7.83e15T^{2} \)
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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

−15.06263025475361286517487267463, −14.16529575921926630719652002680, −12.82436363703067348383073274999, −11.57653046408702644584348665951, −9.657655978883245742518976634365, −8.313082514109970476274144726456, −7.07105734798809488902304062195, −6.34346503936917476332078329627, −4.80069829948449054164389415072, −2.27363017273292034608518438950, 0.40260477446746573872087862472, 1.88422791563667698349543030388, 3.83059422191901187196442584606, 4.64148954479795631686176841437, 7.15892854099430990370142234741, 9.148776513030406393875024601663, 10.05182263132382033705273548638, 10.92523734758186824055592723195, 12.25406727748817378873785835506, 13.07522236012820334962799138191

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