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Properties

Label 2-5-5.3-c6-0-0
Degree $2$
Conductor $5$
Sign $-0.234 - 0.972i$
Analytic cond. $1.15027$
Root an. cond. $1.07250$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−9.58 + 9.58i)2-s + (14.5 + 14.5i)3-s − 119. i·4-s + (88.8 + 87.9i)5-s − 279.·6-s + (59.5 − 59.5i)7-s + (535. + 535. i)8-s − 303. i·9-s + (−1.69e3 + 8.50i)10-s + 800.·11-s + (1.74e3 − 1.74e3i)12-s + (−459. − 459. i)13-s + 1.14e3i·14-s + (12.9 + 2.57e3i)15-s − 2.60e3·16-s + (−2.98e3 + 2.98e3i)17-s + ⋯
L(s)  = 1  + (−1.19 + 1.19i)2-s + (0.540 + 0.540i)3-s − 1.87i·4-s + (0.710 + 0.703i)5-s − 1.29·6-s + (0.173 − 0.173i)7-s + (1.04 + 1.04i)8-s − 0.416i·9-s + (−1.69 + 0.00850i)10-s + 0.601·11-s + (1.01 − 1.01i)12-s + (−0.209 − 0.209i)13-s + 0.416i·14-s + (0.00383 + 0.764i)15-s − 0.635·16-s + (−0.608 + 0.608i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.234 - 0.972i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.234 - 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5\)
Sign: $-0.234 - 0.972i$
Analytic conductor: \(1.15027\)
Root analytic conductor: \(1.07250\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{5} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 5,\ (\ :3),\ -0.234 - 0.972i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.478204 + 0.607363i\)
\(L(\frac12)\) \(\approx\) \(0.478204 + 0.607363i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-88.8 - 87.9i)T \)
good2 \( 1 + (9.58 - 9.58i)T - 64iT^{2} \)
3 \( 1 + (-14.5 - 14.5i)T + 729iT^{2} \)
7 \( 1 + (-59.5 + 59.5i)T - 1.17e5iT^{2} \)
11 \( 1 - 800.T + 1.77e6T^{2} \)
13 \( 1 + (459. + 459. i)T + 4.82e6iT^{2} \)
17 \( 1 + (2.98e3 - 2.98e3i)T - 2.41e7iT^{2} \)
19 \( 1 + 6.97e3iT - 4.70e7T^{2} \)
23 \( 1 + (1.20e4 + 1.20e4i)T + 1.48e8iT^{2} \)
29 \( 1 - 2.63e4iT - 5.94e8T^{2} \)
31 \( 1 - 1.61e4T + 8.87e8T^{2} \)
37 \( 1 + (-1.68e4 + 1.68e4i)T - 2.56e9iT^{2} \)
41 \( 1 + 1.24e5T + 4.75e9T^{2} \)
43 \( 1 + (2.96e4 + 2.96e4i)T + 6.32e9iT^{2} \)
47 \( 1 + (-3.57e4 + 3.57e4i)T - 1.07e10iT^{2} \)
53 \( 1 + (-4.55e4 - 4.55e4i)T + 2.21e10iT^{2} \)
59 \( 1 + 5.23e3iT - 4.21e10T^{2} \)
61 \( 1 - 2.00e4T + 5.15e10T^{2} \)
67 \( 1 + (2.28e5 - 2.28e5i)T - 9.04e10iT^{2} \)
71 \( 1 - 4.71e5T + 1.28e11T^{2} \)
73 \( 1 + (2.42e4 + 2.42e4i)T + 1.51e11iT^{2} \)
79 \( 1 - 4.76e5iT - 2.43e11T^{2} \)
83 \( 1 + (2.96e5 + 2.96e5i)T + 3.26e11iT^{2} \)
89 \( 1 + 8.34e5iT - 4.96e11T^{2} \)
97 \( 1 + (6.03e4 - 6.03e4i)T - 8.32e11iT^{2} \)
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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

−23.99360758855469315074476198734, −21.92626434355142500670309075951, −19.96003259981556314722544271414, −18.25458548985594139704269206635, −17.17094672313422933601882307666, −15.39290304387960786084600832959, −14.33810302360251065432514538488, −10.24420784927499549904886608644, −8.851232586766487961827105606129, −6.61967743645207152387375531906, 1.85368698772543337789492702635, 8.284269800758333109038994109596, 9.769175835864605702233280154032, 11.92082504191767210389529089573, 13.61174178667615861659840104661, 16.82695360168879587627091321153, 18.16893276045428751513998969749, 19.48628972706016732929349056213, 20.51529701941591028045363896621, 21.84453100163284345272516644570

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