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

Degree: \(2\)
Conductor: \(5\)
Sign: $0.506 + 0.862i$
Motivic weight: \(8\)
Character: $\chi_{5} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 5,\ (\ :4),\ 0.506 + 0.862i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(1.20546 - 0.690118i\)
\(L(\frac12)\) \(\approx\) \(1.20546 - 0.690118i\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$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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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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