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Properties

Label 2-490-5.4-c3-0-44
Degree $2$
Conductor $490$
Sign $0.849 + 0.528i$
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 − 1.64i·3-s − 4·4-s + (5.90 − 9.49i)5-s + 3.28·6-s − 8i·8-s + 24.3·9-s + (18.9 + 11.8i)10-s + 45.9·11-s + 6.57i·12-s + 18.6i·13-s + (−15.6 − 9.69i)15-s + 16·16-s − 105. i·17-s + 48.6i·18-s − 141.·19-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.316i·3-s − 0.5·4-s + (0.528 − 0.849i)5-s + 0.223·6-s − 0.353i·8-s + 0.900·9-s + (0.600 + 0.373i)10-s + 1.25·11-s + 0.158i·12-s + 0.398i·13-s + (−0.268 − 0.166i)15-s + 0.250·16-s − 1.50i·17-s + 0.636i·18-s − 1.70·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.849 + 0.528i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.849 + 0.528i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(490\)    =    \(2 \cdot 5 \cdot 7^{2}\)
Sign: $0.849 + 0.528i$
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: \((2,\ 490,\ (\ :3/2),\ 0.849 + 0.528i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.186582209\)
\(L(\frac12)\) \(\approx\) \(2.186582209\)
\(L(\frac{5}{2})\) 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 \( 1 + (-5.90 + 9.49i)T \)
7 \( 1 \)
good3 \( 1 + 1.64iT - 27T^{2} \)
11 \( 1 - 45.9T + 1.33e3T^{2} \)
13 \( 1 - 18.6iT - 2.19e3T^{2} \)
17 \( 1 + 105. iT - 4.91e3T^{2} \)
19 \( 1 + 141.T + 6.85e3T^{2} \)
23 \( 1 - 155. iT - 1.21e4T^{2} \)
29 \( 1 - 90.6T + 2.43e4T^{2} \)
31 \( 1 - 82.2T + 2.97e4T^{2} \)
37 \( 1 + 289. iT - 5.06e4T^{2} \)
41 \( 1 - 326.T + 6.89e4T^{2} \)
43 \( 1 + 78.4iT - 7.95e4T^{2} \)
47 \( 1 + 227. iT - 1.03e5T^{2} \)
53 \( 1 + 10.1iT - 1.48e5T^{2} \)
59 \( 1 - 829.T + 2.05e5T^{2} \)
61 \( 1 + 661.T + 2.26e5T^{2} \)
67 \( 1 + 35.1iT - 3.00e5T^{2} \)
71 \( 1 + 138.T + 3.57e5T^{2} \)
73 \( 1 + 380. iT - 3.89e5T^{2} \)
79 \( 1 - 236.T + 4.93e5T^{2} \)
83 \( 1 + 890. iT - 5.71e5T^{2} \)
89 \( 1 + 121.T + 7.04e5T^{2} \)
97 \( 1 + 1.45e3iT - 9.12e5T^{2} \)
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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

−10.15921643838720512934347604537, −9.295339923116210921534702346811, −8.831173010127126075809207209294, −7.58924136377226826299539327638, −6.77784842151279404861369597905, −5.95661883573465264932612841744, −4.76009414790253347231361128096, −4.01046367071860484161149732314, −1.96862873614981968700725332425, −0.77133394479741909971417816891, 1.33268913680590319557057098993, 2.47835657010464967704886602577, 3.82472339350675036127925673657, 4.50068652512033189079854664686, 6.18024058935266519020625938970, 6.66198637716465837238085067368, 8.152274237438768481841787235853, 9.044801854291189894555229072536, 10.11668498401889832753290012977, 10.45589896358806767462890358928

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