Properties

Degree 2
Conductor 43
Sign $0.934 + 0.357i$
Motivic weight 8
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 17.7i·2-s + 152. i·3-s − 59.7·4-s − 783. i·5-s + 2.70e3·6-s + 1.14e3i·7-s − 3.48e3i·8-s − 1.66e4·9-s − 1.39e4·10-s + 2.13e4·11-s − 9.11e3i·12-s + 1.69e4·13-s + 2.04e4·14-s + 1.19e5·15-s − 7.72e4·16-s + 1.04e5·17-s + ⋯
L(s)  = 1  − 1.11i·2-s + 1.88i·3-s − 0.233·4-s − 1.25i·5-s + 2.09·6-s + 0.478i·7-s − 0.851i·8-s − 2.54·9-s − 1.39·10-s + 1.45·11-s − 0.439i·12-s + 0.593·13-s + 0.531·14-s + 2.35·15-s − 1.17·16-s + 1.25·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.934 + 0.357i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.934 + 0.357i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $0.934 + 0.357i$
motivic weight  =  \(8\)
character  :  $\chi_{43} (42, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 43,\ (\ :4),\ 0.934 + 0.357i)\)
\(L(\frac{9}{2})\)  \(\approx\)  \(2.15021 - 0.396953i\)
\(L(\frac12)\)  \(\approx\)  \(2.15021 - 0.396953i\)
\(L(5)\)   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 \( 1 + (-3.19e6 - 1.22e6i)T \)
good2 \( 1 + 17.7iT - 256T^{2} \)
3 \( 1 - 152. iT - 6.56e3T^{2} \)
5 \( 1 + 783. iT - 3.90e5T^{2} \)
7 \( 1 - 1.14e3iT - 5.76e6T^{2} \)
11 \( 1 - 2.13e4T + 2.14e8T^{2} \)
13 \( 1 - 1.69e4T + 8.15e8T^{2} \)
17 \( 1 - 1.04e5T + 6.97e9T^{2} \)
19 \( 1 - 1.42e5iT - 1.69e10T^{2} \)
23 \( 1 - 3.76e5T + 7.83e10T^{2} \)
29 \( 1 - 2.45e5iT - 5.00e11T^{2} \)
31 \( 1 - 3.73e5T + 8.52e11T^{2} \)
37 \( 1 + 1.72e6iT - 3.51e12T^{2} \)
41 \( 1 + 3.57e6T + 7.98e12T^{2} \)
47 \( 1 - 6.89e6T + 2.38e13T^{2} \)
53 \( 1 + 3.52e6T + 6.22e13T^{2} \)
59 \( 1 + 1.55e6T + 1.46e14T^{2} \)
61 \( 1 + 3.86e6iT - 1.91e14T^{2} \)
67 \( 1 + 1.77e7T + 4.06e14T^{2} \)
71 \( 1 + 3.92e7iT - 6.45e14T^{2} \)
73 \( 1 - 2.65e7iT - 8.06e14T^{2} \)
79 \( 1 + 6.70e7T + 1.51e15T^{2} \)
83 \( 1 + 1.73e7T + 2.25e15T^{2} \)
89 \( 1 + 1.06e7iT - 3.93e15T^{2} \)
97 \( 1 + 1.23e8T + 7.83e15T^{2} \)
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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

−14.25431384180351536379978983920, −12.43102473160243337103342257092, −11.64235381387112878653906933526, −10.48326216163561258176849596688, −9.387115976116497618515946324363, −8.797110379706876942286827767675, −5.68328081834471658282372555570, −4.29637092651803126262031282345, −3.34385661622044802702724956537, −1.14107291358834451033293268753, 1.16312553609002665318239927140, 2.88201175902975034400509664094, 5.96113233723408317959550235476, 6.90020924218322978542255412606, 7.26417471491633468303183282689, 8.652583545725139253680899484418, 11.05928643865594972105803760425, 11.93311426541849231734685218000, 13.57842939822972120197919504349, 14.22577280516725884858979106423

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