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Properties

Label 2-490-5.4-c3-0-56
Degree $2$
Conductor $490$
Sign $-0.829 - 0.558i$
Analytic cond. $28.9109$
Root an. cond. $5.37688$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2i·2-s − 6.44i·3-s − 4·4-s + (6.24 − 9.27i)5-s − 12.8·6-s + 8i·8-s − 14.5·9-s + (−18.5 − 12.4i)10-s + 48.3·11-s + 25.7i·12-s − 93.4i·13-s + (−59.7 − 40.2i)15-s + 16·16-s − 20.2i·17-s + 29.0i·18-s − 31.0·19-s + ⋯
L(s)  = 1  − 0.707i·2-s − 1.23i·3-s − 0.5·4-s + (0.558 − 0.829i)5-s − 0.876·6-s + 0.353i·8-s − 0.537·9-s + (−0.586 − 0.395i)10-s + 1.32·11-s + 0.619i·12-s − 1.99i·13-s + (−1.02 − 0.692i)15-s + 0.250·16-s − 0.289i·17-s + 0.379i·18-s − 0.375·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.829 - 0.558i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.829 - 0.558i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(490\)    =    \(2 \cdot 5 \cdot 7^{2}\)
Sign: $-0.829 - 0.558i$
Analytic conductor: \(28.9109\)
Root analytic conductor: \(5.37688\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{490} (99, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 490,\ (\ :3/2),\ -0.829 - 0.558i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.999334750\)
\(L(\frac12)\) \(\approx\) \(1.999334750\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 2iT \)
5 \( 1 + (-6.24 + 9.27i)T \)
7 \( 1 \)
good3 \( 1 + 6.44iT - 27T^{2} \)
11 \( 1 - 48.3T + 1.33e3T^{2} \)
13 \( 1 + 93.4iT - 2.19e3T^{2} \)
17 \( 1 + 20.2iT - 4.91e3T^{2} \)
19 \( 1 + 31.0T + 6.85e3T^{2} \)
23 \( 1 - 21.0iT - 1.21e4T^{2} \)
29 \( 1 + 69.5T + 2.43e4T^{2} \)
31 \( 1 + 161.T + 2.97e4T^{2} \)
37 \( 1 - 162. iT - 5.06e4T^{2} \)
41 \( 1 - 365.T + 6.89e4T^{2} \)
43 \( 1 + 254. iT - 7.95e4T^{2} \)
47 \( 1 - 468. iT - 1.03e5T^{2} \)
53 \( 1 - 587. iT - 1.48e5T^{2} \)
59 \( 1 - 536.T + 2.05e5T^{2} \)
61 \( 1 - 625.T + 2.26e5T^{2} \)
67 \( 1 + 123. iT - 3.00e5T^{2} \)
71 \( 1 - 210.T + 3.57e5T^{2} \)
73 \( 1 - 141. iT - 3.89e5T^{2} \)
79 \( 1 + 513.T + 4.93e5T^{2} \)
83 \( 1 - 117. iT - 5.71e5T^{2} \)
89 \( 1 + 61.2T + 7.04e5T^{2} \)
97 \( 1 + 436. iT - 9.12e5T^{2} \)
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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

−9.992152658854436608437953061857, −9.140106234229172003812647480153, −8.270044724902960801550921159148, −7.40197663654124739607069966272, −6.18052346303482138413016586463, −5.38847387682935161187709268482, −4.02226726136669864693016492143, −2.59259995573260188063189847685, −1.39928061057943114043727316991, −0.67619972681043237198559733740, 1.90563227413817804401340910549, 3.75439826469045187555034084808, 4.22659100239718058575113481262, 5.51019771244215966514789204006, 6.56865771754298278116915604555, 7.08228346222908993766971139899, 8.722166839982255057169759350763, 9.378182818009816924278441634188, 9.878765184745065305447775510065, 10.99773035571561987769411784672

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