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Properties

Label 2-5-5.3-c20-0-1
Degree $2$
Conductor $5$
Sign $0.0909 - 0.995i$
Analytic cond. $12.6756$
Root an. cond. $3.56029$
Motivic weight $20$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (569. − 569. i)2-s + (−5.61e4 − 5.61e4i)3-s + 4.00e5i·4-s + (−4.35e6 − 8.73e6i)5-s − 6.39e7·6-s + (−1.35e8 + 1.35e8i)7-s + (8.24e8 + 8.24e8i)8-s + 2.82e9i·9-s + (−7.45e9 − 2.49e9i)10-s − 4.76e9·11-s + (2.25e10 − 2.25e10i)12-s + (1.26e11 + 1.26e11i)13-s + 1.53e11i·14-s + (−2.46e11 + 7.35e11i)15-s + 5.18e11·16-s + (−2.33e12 + 2.33e12i)17-s + ⋯
L(s)  = 1  + (0.555 − 0.555i)2-s + (−0.951 − 0.951i)3-s + 0.382i·4-s + (−0.446 − 0.894i)5-s − 1.05·6-s + (−0.478 + 0.478i)7-s + (0.768 + 0.768i)8-s + 0.810i·9-s + (−0.745 − 0.249i)10-s − 0.183·11-s + (0.363 − 0.363i)12-s + (0.914 + 0.914i)13-s + 0.532i·14-s + (−0.426 + 1.27i)15-s + 0.471·16-s + (−1.15 + 1.15i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0909 - 0.995i)\, \overline{\Lambda}(21-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+10) \, L(s)\cr =\mathstrut & (0.0909 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5\)
Sign: $0.0909 - 0.995i$
Analytic conductor: \(12.6756\)
Root analytic conductor: \(3.56029\)
Motivic weight: \(20\)
Rational: no
Arithmetic: yes
Character: $\chi_{5} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 5,\ (\ :10),\ 0.0909 - 0.995i)\)

Particular Values

\(L(\frac{21}{2})\) \(\approx\) \(0.270356 + 0.246801i\)
\(L(\frac12)\) \(\approx\) \(0.270356 + 0.246801i\)
\(L(11)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (4.35e6 + 8.73e6i)T \)
good2 \( 1 + (-569. + 569. i)T - 1.04e6iT^{2} \)
3 \( 1 + (5.61e4 + 5.61e4i)T + 3.48e9iT^{2} \)
7 \( 1 + (1.35e8 - 1.35e8i)T - 7.97e16iT^{2} \)
11 \( 1 + 4.76e9T + 6.72e20T^{2} \)
13 \( 1 + (-1.26e11 - 1.26e11i)T + 1.90e22iT^{2} \)
17 \( 1 + (2.33e12 - 2.33e12i)T - 4.06e24iT^{2} \)
19 \( 1 - 1.66e12iT - 3.75e25T^{2} \)
23 \( 1 + (4.61e13 + 4.61e13i)T + 1.71e27iT^{2} \)
29 \( 1 + 5.94e14iT - 1.76e29T^{2} \)
31 \( 1 + 3.56e13T + 6.71e29T^{2} \)
37 \( 1 + (2.17e15 - 2.17e15i)T - 2.31e31iT^{2} \)
41 \( 1 + 1.52e16T + 1.80e32T^{2} \)
43 \( 1 + (-1.84e16 - 1.84e16i)T + 4.67e32iT^{2} \)
47 \( 1 + (3.49e16 - 3.49e16i)T - 2.76e33iT^{2} \)
53 \( 1 + (4.97e16 + 4.97e16i)T + 3.05e34iT^{2} \)
59 \( 1 + 2.17e17iT - 2.61e35T^{2} \)
61 \( 1 - 9.88e17T + 5.08e35T^{2} \)
67 \( 1 + (1.74e17 - 1.74e17i)T - 3.32e36iT^{2} \)
71 \( 1 + 2.00e18T + 1.05e37T^{2} \)
73 \( 1 + (-1.27e18 - 1.27e18i)T + 1.84e37iT^{2} \)
79 \( 1 + 1.35e19iT - 8.96e37T^{2} \)
83 \( 1 + (1.95e19 + 1.95e19i)T + 2.40e38iT^{2} \)
89 \( 1 - 2.33e19iT - 9.72e38T^{2} \)
97 \( 1 + (3.66e19 - 3.66e19i)T - 5.43e39iT^{2} \)
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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

−19.09190924281341830163212288236, −17.44394309334606661361887688118, −16.15994095579909424798727868554, −13.27574734352455311447560023493, −12.39174665732862990132694995090, −11.38912365688685624532886428643, −8.344274117820672490551266809543, −6.22344108076549577066894726026, −4.21482481640528147620939297710, −1.77561223380849530413802737923, 0.14531743132646571948372583694, 3.80804601653695376664672847659, 5.42131233628814040798337332862, 6.89371008727910873807526951106, 10.15009458690231309249213830903, 11.13468943295489290646557525025, 13.67663575720616153180212053732, 15.47930938120862158038529306989, 16.05766497403852984865237740934, 18.06118451283362741528017429167

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