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Properties

Label 2-490-35.12-c1-0-16
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: $\chi_{490} (117, \cdot )$
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

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

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