Properties

Degree 2
Conductor 5
Sign $0.506 - 0.862i$
Motivic weight 8
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−4.39 + 4.39i)2-s + (75.2 + 75.2i)3-s + 217. i·4-s + (−14.1 − 624. i)5-s − 662.·6-s + (730. − 730. i)7-s + (−2.08e3 − 2.08e3i)8-s + 4.77e3i·9-s + (2.80e3 + 2.68e3i)10-s + 1.95e4·11-s + (−1.63e4 + 1.63e4i)12-s + (−2.49e4 − 2.49e4i)13-s + 6.42e3i·14-s + (4.59e4 − 4.81e4i)15-s − 3.73e4·16-s + (−1.12e4 + 1.12e4i)17-s + ⋯
L(s)  = 1  + (−0.274 + 0.274i)2-s + (0.929 + 0.929i)3-s + 0.849i·4-s + (−0.0226 − 0.999i)5-s − 0.510·6-s + (0.304 − 0.304i)7-s + (−0.508 − 0.508i)8-s + 0.728i·9-s + (0.280 + 0.268i)10-s + 1.33·11-s + (−0.789 + 0.789i)12-s + (−0.872 − 0.872i)13-s + 0.167i·14-s + (0.908 − 0.950i)15-s − 0.569·16-s + (−0.135 + 0.135i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.506 - 0.862i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.506 - 0.862i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(5\)
\( \varepsilon \)  =  $0.506 - 0.862i$
motivic weight  =  \(8\)
character  :  $\chi_{5} (3, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 5,\ (\ :4),\ 0.506 - 0.862i)\)
\(L(\frac{9}{2})\)  \(\approx\)  \(1.20546 + 0.690118i\)
\(L(\frac12)\)  \(\approx\)  \(1.20546 + 0.690118i\)
\(L(5)\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \neq 5$,\(F_p(T)\) is a polynomial of degree 2. If $p = 5$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad5 \( 1 + (14.1 + 624. i)T \)
good2 \( 1 + (4.39 - 4.39i)T - 256iT^{2} \)
3 \( 1 + (-75.2 - 75.2i)T + 6.56e3iT^{2} \)
7 \( 1 + (-730. + 730. i)T - 5.76e6iT^{2} \)
11 \( 1 - 1.95e4T + 2.14e8T^{2} \)
13 \( 1 + (2.49e4 + 2.49e4i)T + 8.15e8iT^{2} \)
17 \( 1 + (1.12e4 - 1.12e4i)T - 6.97e9iT^{2} \)
19 \( 1 - 1.71e5iT - 1.69e10T^{2} \)
23 \( 1 + (1.32e5 + 1.32e5i)T + 7.83e10iT^{2} \)
29 \( 1 - 1.27e5iT - 5.00e11T^{2} \)
31 \( 1 + 9.60e5T + 8.52e11T^{2} \)
37 \( 1 + (2.43e5 - 2.43e5i)T - 3.51e12iT^{2} \)
41 \( 1 - 2.50e6T + 7.98e12T^{2} \)
43 \( 1 + (6.76e3 + 6.76e3i)T + 1.16e13iT^{2} \)
47 \( 1 + (1.79e6 - 1.79e6i)T - 2.38e13iT^{2} \)
53 \( 1 + (2.97e6 + 2.97e6i)T + 6.22e13iT^{2} \)
59 \( 1 + 3.13e5iT - 1.46e14T^{2} \)
61 \( 1 - 1.76e7T + 1.91e14T^{2} \)
67 \( 1 + (-4.41e6 + 4.41e6i)T - 4.06e14iT^{2} \)
71 \( 1 + 8.89e6T + 6.45e14T^{2} \)
73 \( 1 + (-1.95e7 - 1.95e7i)T + 8.06e14iT^{2} \)
79 \( 1 - 1.11e7iT - 1.51e15T^{2} \)
83 \( 1 + (-1.58e7 - 1.58e7i)T + 2.25e15iT^{2} \)
89 \( 1 + 4.85e7iT - 3.93e15T^{2} \)
97 \( 1 + (1.07e8 - 1.07e8i)T - 7.83e15iT^{2} \)
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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

−22.15143146549113656787806301125, −20.84056587771569478031415237667, −19.95342910457639433500787954293, −17.33838497268545920235751678290, −16.20632418223863919495644201406, −14.53044912107481401064612306396, −12.40835243697841491853441164719, −9.481800254373481265993199787565, −8.153015598397907333233809721619, −3.97762592032845800453370154856, 2.09579531197545286769574704115, 6.90603413569605103844745153203, 9.258610739881257621293271378059, 11.49882662871319202047641892491, 14.01672240417674220636121976109, 14.83531293857256237771010437010, 17.92358357817543019865050227491, 19.19929867529326751825832684623, 19.77320222417743567463796232115, 22.03491082930842241619664755591

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