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Properties

Label 2-5-5.4-c17-0-4
Degree $2$
Conductor $5$
Sign $0.866 + 0.499i$
Analytic cond. $9.16110$
Root an. cond. $3.02673$
Motivic weight $17$
Arithmetic yes
Rational no
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

Degree: \(2\)
Conductor: \(5\)
Sign: $0.866 + 0.499i$
Analytic conductor: \(9.16110\)
Root analytic conductor: \(3.02673\)
Motivic weight: \(17\)
Rational: no
Arithmetic: yes
Character: $\chi_{5} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 5,\ (\ :17/2),\ 0.866 + 0.499i)\)

Particular Values

\(L(9)\) \(\approx\) \(1.71357 - 0.458674i\)
\(L(\frac12)\) \(\approx\) \(1.71357 - 0.458674i\)
\(L(\frac{19}{2})\) not available
\(L(1)\) not available

Euler product

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

Imaginary part of the first few zeros on the critical line

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

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