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Properties

Label 2-490-35.3-c1-0-8
Degree $2$
Conductor $490$
Sign $-0.0254 - 0.999i$
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 + (0.752 + 2.80i)3-s + (0.866 + 0.499i)4-s + (1.38 − 1.75i)5-s + 2.90i·6-s + (0.707 + 0.707i)8-s + (−4.71 + 2.72i)9-s + (1.79 − 1.33i)10-s + (−1.83 + 3.17i)11-s + (−0.752 + 2.80i)12-s + (0.830 − 0.830i)13-s + (5.97 + 2.55i)15-s + (0.500 + 0.866i)16-s + (0.761 − 0.204i)17-s + (−5.26 + 1.41i)18-s + (1.09 + 1.89i)19-s + ⋯
L(s)  = 1  + (0.683 + 0.183i)2-s + (0.434 + 1.62i)3-s + (0.433 + 0.249i)4-s + (0.618 − 0.785i)5-s + 1.18i·6-s + (0.249 + 0.249i)8-s + (−1.57 + 0.908i)9-s + (0.566 − 0.423i)10-s + (−0.553 + 0.958i)11-s + (−0.217 + 0.810i)12-s + (0.230 − 0.230i)13-s + (1.54 + 0.660i)15-s + (0.125 + 0.216i)16-s + (0.184 − 0.0494i)17-s + (−1.24 + 0.332i)18-s + (0.251 + 0.434i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0254 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0254 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(490\)    =    \(2 \cdot 5 \cdot 7^{2}\)
Sign: $-0.0254 - 0.999i$
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: \((2,\ 490,\ (\ :1/2),\ -0.0254 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.77745 + 1.82321i\)
\(L(\frac12)\) \(\approx\) \(1.77745 + 1.82321i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.38 + 1.75i)T \)
7 \( 1 \)
good3 \( 1 + (-0.752 - 2.80i)T + (-2.59 + 1.5i)T^{2} \)
11 \( 1 + (1.83 - 3.17i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.830 + 0.830i)T - 13iT^{2} \)
17 \( 1 + (-0.761 + 0.204i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (-1.09 - 1.89i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.21 + 4.54i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + 2.62iT - 29T^{2} \)
31 \( 1 + (0.0359 + 0.0207i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.248 + 0.0664i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + 8.98iT - 41T^{2} \)
43 \( 1 + (0.474 + 0.474i)T + 43iT^{2} \)
47 \( 1 + (1.65 - 6.18i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (7.64 - 2.04i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (-5.35 + 9.27i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.72 - 0.996i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.71 + 6.39i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 - 8.11T + 71T^{2} \)
73 \( 1 + (2.55 + 9.52i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (-11.6 + 6.70i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (9.73 - 9.73i)T - 83iT^{2} \)
89 \( 1 + (-0.715 - 1.23i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (3.16 + 3.16i)T + 97iT^{2} \)
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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

−10.94825268583214372134098309395, −10.19650152515861124667044616258, −9.545165436977305670956680223245, −8.669262198306337866318876715718, −7.73527764543040198346517542834, −6.17420689681925090866133295242, −5.11850405394936185739097860577, −4.65097913932593246754349385804, −3.59204276385765474332980150597, −2.30924569077700107711139118147, 1.40255704264584058184713498144, 2.61110030797715758726063930987, 3.36518681640552734961862460843, 5.37147549023091018925517283741, 6.20194481638864252291085464844, 6.94753828619265599565541090240, 7.74234301330093569819717266994, 8.752475418604755164461339723779, 9.956017350879075320795592501555, 11.13864813417088438747707052278

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