Properties

Degree 2
Conductor 5
Sign $0.866 - 0.499i$
Motivic weight 17
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 275. i·2-s + 2.99e3i·3-s + 5.52e4·4-s + (−7.56e5 + 4.36e5i)5-s + 8.23e5·6-s + 2.43e7i·7-s − 5.13e7i·8-s + 1.20e8·9-s + (1.20e8 + 2.08e8i)10-s + 8.41e8·11-s + 1.65e8i·12-s + 4.01e9i·13-s + 6.71e9·14-s + (−1.30e9 − 2.26e9i)15-s − 6.88e9·16-s − 8.60e9i·17-s + ⋯
L(s)  = 1  − 0.760i·2-s + 0.263i·3-s + 0.421·4-s + (−0.866 + 0.499i)5-s + 0.200·6-s + 1.59i·7-s − 1.08i·8-s + 0.930·9-s + (0.379 + 0.658i)10-s + 1.18·11-s + 0.110i·12-s + 1.36i·13-s + 1.21·14-s + (−0.131 − 0.227i)15-s − 0.400·16-s − 0.299i·17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.866 - 0.499i)\, \overline{\Lambda}(18-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & (0.866 - 0.499i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(5\)
\( \varepsilon \)  =  $0.866 - 0.499i$
motivic weight  =  \(17\)
character  :  $\chi_{5} (4, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 5,\ (\ :17/2),\ 0.866 - 0.499i)$
$L(9)$  $\approx$  $1.71357 + 0.458674i$
$L(\frac12)$  $\approx$  $1.71357 + 0.458674i$
$L(\frac{19}{2})$   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 \( 1 + (7.56e5 - 4.36e5i)T \)
good2 \( 1 + 275. iT - 1.31e5T^{2} \)
3 \( 1 - 2.99e3iT - 1.29e8T^{2} \)
7 \( 1 - 2.43e7iT - 2.32e14T^{2} \)
11 \( 1 - 8.41e8T + 5.05e17T^{2} \)
13 \( 1 - 4.01e9iT - 8.65e18T^{2} \)
17 \( 1 + 8.60e9iT - 8.27e20T^{2} \)
19 \( 1 + 6.64e10T + 5.48e21T^{2} \)
23 \( 1 - 5.12e11iT - 1.41e23T^{2} \)
29 \( 1 - 1.63e11T + 7.25e24T^{2} \)
31 \( 1 + 3.12e12T + 2.25e25T^{2} \)
37 \( 1 - 2.41e12iT - 4.56e26T^{2} \)
41 \( 1 - 4.43e13T + 2.61e27T^{2} \)
43 \( 1 + 7.24e13iT - 5.87e27T^{2} \)
47 \( 1 + 5.71e13iT - 2.66e28T^{2} \)
53 \( 1 + 3.35e14iT - 2.05e29T^{2} \)
59 \( 1 + 2.43e14T + 1.27e30T^{2} \)
61 \( 1 - 2.41e15T + 2.24e30T^{2} \)
67 \( 1 + 2.63e15iT - 1.10e31T^{2} \)
71 \( 1 - 4.51e15T + 2.96e31T^{2} \)
73 \( 1 + 2.61e15iT - 4.74e31T^{2} \)
79 \( 1 - 7.48e15T + 1.81e32T^{2} \)
83 \( 1 + 2.07e15iT - 4.21e32T^{2} \)
89 \( 1 - 2.77e15T + 1.37e33T^{2} \)
97 \( 1 + 5.77e16iT - 5.95e33T^{2} \)
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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

−19.43703030390757734195865004946, −18.72420707379999555079285017828, −16.06798904092116688287132404603, −14.99608921421173750699860116336, −12.23576327053363562535055148682, −11.35730230984417893877341977366, −9.323204538467127664011854347828, −6.77381539174281470743275851829, −3.83868910019167239092593741020, −1.94433282333420287482938492760, 0.944887904975419082232510492960, 4.16927888261452006715223820268, 6.79752005992018446053471276608, 7.967273332553360160275088699638, 10.76144521427401133977686507707, 12.72060941111265965618889592093, 14.72539605578612292571258392966, 16.22591451468609401219458545385, 17.27027385942891471962120152446, 19.62366270359354639673658785765

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