↑

Properties

Label 2-490-7.4-c1-0-5
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

Related objects

Downloads

Learn more

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: $\chi_{490} (361, \cdot )$
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 \( 1 - 16.2T + 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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