Properties

Label 2-490-7.2-c1-0-7
Degree $2$
Conductor $490$
Sign $-0.0725 + 0.997i$
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  + (0.5 − 0.866i)2-s + (0.292 + 0.507i)3-s + (−0.499 − 0.866i)4-s + (0.5 − 0.866i)5-s + 0.585·6-s − 0.999·8-s + (1.32 − 2.30i)9-s + (−0.499 − 0.866i)10-s + (−2.41 − 4.18i)11-s + (0.292 − 0.507i)12-s − 0.828·13-s + 0.585·15-s + (−0.5 + 0.866i)16-s + (2.70 + 4.68i)17-s + (−1.32 − 2.30i)18-s + (1.70 − 2.95i)19-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (0.169 + 0.292i)3-s + (−0.249 − 0.433i)4-s + (0.223 − 0.387i)5-s + 0.239·6-s − 0.353·8-s + (0.442 − 0.766i)9-s + (−0.158 − 0.273i)10-s + (−0.727 − 1.26i)11-s + (0.0845 − 0.146i)12-s − 0.229·13-s + 0.151·15-s + (−0.125 + 0.216i)16-s + (0.656 + 1.13i)17-s + (−0.313 − 0.542i)18-s + (0.391 − 0.678i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0725 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0725 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(490\)    =    \(2 \cdot 5 \cdot 7^{2}\)
Sign: $-0.0725 + 0.997i$
Analytic conductor: \(3.91266\)
Root analytic conductor: \(1.97804\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{490} (471, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 490,\ (\ :1/2),\ -0.0725 + 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.18336 - 1.27258i\)
\(L(\frac12)\) \(\approx\) \(1.18336 - 1.27258i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 + 0.866i)T \)
5 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 \)
good3 \( 1 + (-0.292 - 0.507i)T + (-1.5 + 2.59i)T^{2} \)
11 \( 1 + (2.41 + 4.18i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 0.828T + 13T^{2} \)
17 \( 1 + (-2.70 - 4.68i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.70 + 2.95i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.41 + 5.91i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 0.828T + 29T^{2} \)
31 \( 1 + (-1.41 - 2.44i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.82 - 3.16i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 11.0T + 41T^{2} \)
43 \( 1 + 3.17T + 43T^{2} \)
47 \( 1 + (-5.41 + 9.37i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-5.24 - 9.08i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-5.70 - 9.88i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (6.65 - 11.5i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4.82 - 8.36i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 12.4T + 71T^{2} \)
73 \( 1 + (3.29 + 5.70i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.585 + 1.01i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 6.24T + 83T^{2} \)
89 \( 1 + (6.36 - 11.0i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 16.2T + 97T^{2} \)
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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

−10.48600277224350097108032493380, −10.19008858846296134917408832075, −8.893655795852875580180685789277, −8.444630176124162426197085404849, −6.91020201690526679778495290001, −5.82452444109223690570197784781, −4.90794096059190083941813958921, −3.74370077305351233896839344449, −2.78099881698057409319702680027, −0.993803227304078733629208062623, 2.01031083241166446899231275834, 3.29374257017834999970556924054, 4.82692287645340763371641627912, 5.40106594110054251686578017732, 6.87230358838467324004666440180, 7.42916779155024807746476905443, 8.097308375099413384463413235554, 9.588457186991777021647955557611, 10.05483588893447218573185654120, 11.26916214430245628482955299629

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