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Properties

Label 2-490-35.4-c1-0-15
Degree $2$
Conductor $490$
Sign $0.878 + 0.477i$
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  + (0.866 − 0.5i)2-s + (2.12 + 1.22i)3-s + (0.499 − 0.866i)4-s + (−0.917 − 2.03i)5-s + 2.44·6-s − 0.999i·8-s + (1.49 + 2.59i)9-s + (−1.81 − 1.30i)10-s + (2.44 − 4.24i)11-s + (2.12 − 1.22i)12-s + 4.44i·13-s + (0.550 − 5.44i)15-s + (−0.5 − 0.866i)16-s + (1.73 + i)17-s + (2.59 + 1.49i)18-s + (0.775 + 1.34i)19-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (1.22 + 0.707i)3-s + (0.249 − 0.433i)4-s + (−0.410 − 0.911i)5-s + 0.999·6-s − 0.353i·8-s + (0.499 + 0.866i)9-s + (−0.573 − 0.413i)10-s + (0.738 − 1.27i)11-s + (0.612 − 0.353i)12-s + 1.23i·13-s + (0.142 − 1.40i)15-s + (−0.125 − 0.216i)16-s + (0.420 + 0.242i)17-s + (0.612 + 0.353i)18-s + (0.177 + 0.308i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.878 + 0.477i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.878 + 0.477i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(490\)    =    \(2 \cdot 5 \cdot 7^{2}\)
Sign: $0.878 + 0.477i$
Analytic conductor: \(3.91266\)
Root analytic conductor: \(1.97804\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{490} (459, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 490,\ (\ :1/2),\ 0.878 + 0.477i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.68328 - 0.681623i\)
\(L(\frac12)\) \(\approx\) \(2.68328 - 0.681623i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 + 0.5i)T \)
5 \( 1 + (0.917 + 2.03i)T \)
7 \( 1 \)
good3 \( 1 + (-2.12 - 1.22i)T + (1.5 + 2.59i)T^{2} \)
11 \( 1 + (-2.44 + 4.24i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 4.44iT - 13T^{2} \)
17 \( 1 + (-1.73 - i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.775 - 1.34i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.51 + 1.44i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 6.89T + 29T^{2} \)
31 \( 1 + (4.44 - 7.70i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.73 - i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 1.10T + 41T^{2} \)
43 \( 1 + 0.898iT - 43T^{2} \)
47 \( 1 + (7.70 - 4.44i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-9.43 - 5.44i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (0.775 - 1.34i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.77 + 3.07i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.92 + 4i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 1.10T + 71T^{2} \)
73 \( 1 + (2.51 + 1.44i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-3.44 - 5.97i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 2.44iT - 83T^{2} \)
89 \( 1 + (5 + 8.66i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 15.7iT - 97T^{2} \)
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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

−11.02875602679030469064872607705, −9.830927780757886092726297402146, −8.907078779100063074510712040358, −8.690648530564064187029329352494, −7.38606157507819039739889022484, −6.00874753408865563431502639532, −4.81292882519065659206760641238, −3.85892183038002016229836722397, −3.28390938843505061828223949854, −1.59101018622475147399570632424, 2.04943278758552499493824385815, 3.11254873888166014101658308194, 3.93389165222360943651026113466, 5.44348703551328686381103694975, 6.77774829730987391996516273405, 7.42839342556143145616328489391, 7.894812141811293010832440666534, 9.096482217217349710197145371789, 10.03420688366574565003876441387, 11.23333120524485898999231975563

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