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Properties

Label 2-490-35.3-c1-0-19
Degree $2$
Conductor $490$
Sign $0.116 - 0.993i$
Analytic cond. $3.91266$
Root an. cond. $1.97804$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.965 − 0.258i)2-s + (−0.830 − 3.10i)3-s + (0.866 + 0.499i)4-s + (−1.65 − 1.50i)5-s + 3.20i·6-s + (−0.707 − 0.707i)8-s + (−6.32 + 3.65i)9-s + (1.20 + 1.88i)10-s + (−1.22 + 2.12i)11-s + (0.830 − 3.10i)12-s + (0.884 − 0.884i)13-s + (−3.28 + 6.37i)15-s + (0.500 + 0.866i)16-s + (−4.69 + 1.25i)17-s + (7.05 − 1.89i)18-s + (0.522 + 0.904i)19-s + ⋯
L(s)  = 1  + (−0.683 − 0.183i)2-s + (−0.479 − 1.79i)3-s + (0.433 + 0.249i)4-s + (−0.740 − 0.672i)5-s + 1.31i·6-s + (−0.249 − 0.249i)8-s + (−2.10 + 1.21i)9-s + (0.382 + 0.594i)10-s + (−0.370 + 0.641i)11-s + (0.239 − 0.895i)12-s + (0.245 − 0.245i)13-s + (−0.849 + 1.64i)15-s + (0.125 + 0.216i)16-s + (−1.13 + 0.304i)17-s + (1.66 − 0.445i)18-s + (0.119 + 0.207i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.116 - 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.116 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(490\)    =    \(2 \cdot 5 \cdot 7^{2}\)
Sign: $0.116 - 0.993i$
Analytic conductor: \(3.91266\)
Root analytic conductor: \(1.97804\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{490} (423, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 490,\ (\ :1/2),\ 0.116 - 0.993i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0843117 + 0.0749919i\)
\(L(\frac12)\) \(\approx\) \(0.0843117 + 0.0749919i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.965 + 0.258i)T \)
5 \( 1 + (1.65 + 1.50i)T \)
7 \( 1 \)
good3 \( 1 + (0.830 + 3.10i)T + (-2.59 + 1.5i)T^{2} \)
11 \( 1 + (1.22 - 2.12i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.884 + 0.884i)T - 13iT^{2} \)
17 \( 1 + (4.69 - 1.25i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (-0.522 - 0.904i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.13 + 7.95i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 - 1.56iT - 29T^{2} \)
31 \( 1 + (0.594 + 0.343i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.63 + 0.706i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + 7.63iT - 41T^{2} \)
43 \( 1 + (-4.56 - 4.56i)T + 43iT^{2} \)
47 \( 1 + (2.63 - 9.82i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (1.72 - 0.462i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (3.42 - 5.93i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.544 - 0.314i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.136 - 0.507i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + 10.9T + 71T^{2} \)
73 \( 1 + (2.60 + 9.73i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (7.06 - 4.07i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (6.80 - 6.80i)T - 83iT^{2} \)
89 \( 1 + (3.90 + 6.77i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (7.49 + 7.49i)T + 97iT^{2} \)
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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

−10.61191792190084099297563836352, −8.982716293972001231713107159814, −8.353842898432888614893698479179, −7.54716966162613692256195344263, −6.88909428073686888841306377257, −5.89647409286761809262709590493, −4.56417319928610221864711351547, −2.65434961801262101868266494504, −1.42512848179034518771492756270, −0.091965629725217424128822578880, 2.97009881909563903380034569702, 3.90122360014724642038514062684, 5.01685325836112054425908844875, 6.03141725674329556972554487648, 7.11050918290738994925118001749, 8.347951928011509381259211013612, 9.110430361467638261815697600173, 9.929867073197901869395428144798, 10.75553777256778735773663461525, 11.31157056177292098885919906481

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