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Properties

Label 2-490-35.12-c1-0-8
Degree $2$
Conductor $490$
Sign $0.866 + 0.498i$
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 + (0.0749 − 0.279i)3-s + (0.866 − 0.499i)4-s + (−2.20 + 0.378i)5-s + 0.289i·6-s + (−0.707 + 0.707i)8-s + (2.52 + 1.45i)9-s + (2.03 − 0.935i)10-s + (−2.81 − 4.87i)11-s + (−0.0749 − 0.279i)12-s + (1.42 + 1.42i)13-s + (−0.0593 + 0.645i)15-s + (0.500 − 0.866i)16-s + (5.12 + 1.37i)17-s + (−2.81 − 0.754i)18-s + (1.94 − 3.37i)19-s + ⋯
L(s)  = 1  + (−0.683 + 0.183i)2-s + (0.0432 − 0.161i)3-s + (0.433 − 0.249i)4-s + (−0.985 + 0.169i)5-s + 0.118i·6-s + (−0.249 + 0.249i)8-s + (0.841 + 0.486i)9-s + (0.642 − 0.295i)10-s + (−0.848 − 1.46i)11-s + (−0.0216 − 0.0807i)12-s + (0.396 + 0.396i)13-s + (−0.0153 + 0.166i)15-s + (0.125 − 0.216i)16-s + (1.24 + 0.333i)17-s + (−0.663 − 0.177i)18-s + (0.446 − 0.773i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.866 + 0.498i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.866 + 0.498i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(490\)    =    \(2 \cdot 5 \cdot 7^{2}\)
Sign: $0.866 + 0.498i$
Analytic conductor: \(3.91266\)
Root analytic conductor: \(1.97804\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{490} (117, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 490,\ (\ :1/2),\ 0.866 + 0.498i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.869660 - 0.232279i\)
\(L(\frac12)\) \(\approx\) \(0.869660 - 0.232279i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.20 - 0.378i)T \)
7 \( 1 \)
good3 \( 1 + (-0.0749 + 0.279i)T + (-2.59 - 1.5i)T^{2} \)
11 \( 1 + (2.81 + 4.87i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.42 - 1.42i)T + 13iT^{2} \)
17 \( 1 + (-5.12 - 1.37i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (-1.94 + 3.37i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.290 + 1.08i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + 3.15iT - 29T^{2} \)
31 \( 1 + (-3.33 + 1.92i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-4.86 + 1.30i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + 7.21iT - 41T^{2} \)
43 \( 1 + (-1.85 + 1.85i)T - 43iT^{2} \)
47 \( 1 + (-1.52 - 5.69i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-1.33 - 0.357i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (2.73 + 4.74i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.99 - 2.30i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.218 + 0.816i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 - 4.77T + 71T^{2} \)
73 \( 1 + (1.45 - 5.42i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (-5.41 - 3.12i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (5.67 + 5.67i)T + 83iT^{2} \)
89 \( 1 + (-5.96 + 10.3i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-6.63 + 6.63i)T - 97iT^{2} \)
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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

−10.86123071453531015161472372896, −10.10174795254042262849483113588, −8.918163058901148551168353663348, −7.986521337787431474506662330647, −7.60806670184400069739547417928, −6.47090782162897428591482117457, −5.34877193387356075381229127166, −3.97249924408624615873869925807, −2.74614935442284125740697598180, −0.834559785201316134383351470587, 1.24080776639593799184182007511, 3.04936713693857943822813342141, 4.13808506089598480133815537892, 5.25263047522571202428444727148, 6.81640787795945548595688867593, 7.64428090483481430408934919099, 8.140193254179220342337604765023, 9.494052893898828456463316312145, 10.02372664188342833971538040760, 10.84288497214438431818309347428

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