Properties

Degree 2
Conductor 5
Sign $-0.814 + 0.580i$
Motivic weight 9
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 41.3i·2-s + 37.6i·3-s − 1.19e3·4-s + (1.13e3 − 810. i)5-s + 1.55e3·6-s − 5.31e3i·7-s + 2.82e4i·8-s + 1.82e4·9-s + (−3.35e4 − 4.70e4i)10-s + 1.04e4·11-s − 4.49e4i·12-s + 7.96e4i·13-s − 2.19e5·14-s + (3.05e4 + 4.28e4i)15-s + 5.54e5·16-s + 3.13e5i·17-s + ⋯
L(s)  = 1  − 1.82i·2-s + 0.268i·3-s − 2.33·4-s + (0.814 − 0.580i)5-s + 0.489·6-s − 0.836i·7-s + 2.43i·8-s + 0.928·9-s + (−1.05 − 1.48i)10-s + 0.214·11-s − 0.626i·12-s + 0.773i·13-s − 1.52·14-s + (0.155 + 0.218i)15-s + 2.11·16-s + 0.911i·17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.814 + 0.580i)\, \overline{\Lambda}(10-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.814 + 0.580i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(5\)
\( \varepsilon \)  =  $-0.814 + 0.580i$
motivic weight  =  \(9\)
character  :  $\chi_{5} (4, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 5,\ (\ :9/2),\ -0.814 + 0.580i)$
$L(5)$  $\approx$  $0.401675 - 1.25623i$
$L(\frac12)$  $\approx$  $0.401675 - 1.25623i$
$L(\frac{11}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 5$, \(F_p\) is a polynomial of degree 2. If $p = 5$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad5 \( 1 + (-1.13e3 + 810. i)T \)
good2 \( 1 + 41.3iT - 512T^{2} \)
3 \( 1 - 37.6iT - 1.96e4T^{2} \)
7 \( 1 + 5.31e3iT - 4.03e7T^{2} \)
11 \( 1 - 1.04e4T + 2.35e9T^{2} \)
13 \( 1 - 7.96e4iT - 1.06e10T^{2} \)
17 \( 1 - 3.13e5iT - 1.18e11T^{2} \)
19 \( 1 - 2.46e5T + 3.22e11T^{2} \)
23 \( 1 + 7.21e5iT - 1.80e12T^{2} \)
29 \( 1 + 2.56e6T + 1.45e13T^{2} \)
31 \( 1 + 3.29e6T + 2.64e13T^{2} \)
37 \( 1 - 1.40e7iT - 1.29e14T^{2} \)
41 \( 1 - 1.70e7T + 3.27e14T^{2} \)
43 \( 1 + 2.92e7iT - 5.02e14T^{2} \)
47 \( 1 - 4.10e7iT - 1.11e15T^{2} \)
53 \( 1 - 5.67e7iT - 3.29e15T^{2} \)
59 \( 1 + 1.60e8T + 8.66e15T^{2} \)
61 \( 1 - 5.33e7T + 1.16e16T^{2} \)
67 \( 1 + 2.80e8iT - 2.72e16T^{2} \)
71 \( 1 + 8.97e7T + 4.58e16T^{2} \)
73 \( 1 - 7.60e7iT - 5.88e16T^{2} \)
79 \( 1 + 4.10e8T + 1.19e17T^{2} \)
83 \( 1 - 5.21e8iT - 1.86e17T^{2} \)
89 \( 1 + 2.37e8T + 3.50e17T^{2} \)
97 \( 1 + 6.03e8iT - 7.60e17T^{2} \)
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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

−21.15813871430524449751723642620, −20.15969939062536781994329171645, −18.61720263707818649312312787820, −16.99921690732488005546092096965, −13.83244558470436788780220785247, −12.59051006570527268560620301989, −10.60380997660035783241402628780, −9.368755920511361841287164112360, −4.27132325936125202902816454889, −1.44920740204565842402946020490, 5.63865186807939055305227023641, 7.28687580313160429112295426927, 9.437150296272990796069135813970, 13.18334852706229502296228920541, 14.67760821266155244634966781811, 15.96632906290911725740572593786, 17.76341186864503655243786273213, 18.53153659063344673451850935826, 21.72889698142492311186561281409, 22.88370253057141154788389003688

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