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Properties

Label 2-490-7.2-c1-0-2
Degree $2$
Conductor $490$
Sign $0.991 + 0.126i$
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 + (0.5 + 0.866i)3-s + (−0.499 − 0.866i)4-s + (−0.5 + 0.866i)5-s + 0.999·6-s − 0.999·8-s + (1 − 1.73i)9-s + (0.499 + 0.866i)10-s + (3 + 5.19i)11-s + (0.499 − 0.866i)12-s + 4·13-s − 0.999·15-s + (−0.5 + 0.866i)16-s + (−1 − 1.73i)18-s + (1 − 1.73i)19-s + 0.999·20-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (0.288 + 0.499i)3-s + (−0.249 − 0.433i)4-s + (−0.223 + 0.387i)5-s + 0.408·6-s − 0.353·8-s + (0.333 − 0.577i)9-s + (0.158 + 0.273i)10-s + (0.904 + 1.56i)11-s + (0.144 − 0.249i)12-s + 1.10·13-s − 0.258·15-s + (−0.125 + 0.216i)16-s + (−0.235 − 0.408i)18-s + (0.229 − 0.397i)19-s + 0.223·20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 + 0.126i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 + 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(490\)    =    \(2 \cdot 5 \cdot 7^{2}\)
Sign: $0.991 + 0.126i$
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: \((2,\ 490,\ (\ :1/2),\ 0.991 + 0.126i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.91224 - 0.121351i\)
\(L(\frac12)\) \(\approx\) \(1.91224 - 0.121351i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 - 0.866i)T + (-1.5 + 2.59i)T^{2} \)
11 \( 1 + (-3 - 5.19i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 4T + 13T^{2} \)
17 \( 1 + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1 + 1.73i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.5 + 2.59i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 3T + 29T^{2} \)
31 \( 1 + (-4 - 6.92i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2 + 3.46i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 9T + 41T^{2} \)
43 \( 1 + 7T + 43T^{2} \)
47 \( 1 + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-3 - 5.19i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (3 + 5.19i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.5 + 4.33i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.5 + 4.33i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 6T + 71T^{2} \)
73 \( 1 + (8 + 13.8i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (1 - 1.73i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 3T + 83T^{2} \)
89 \( 1 + (7.5 - 12.9i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 14T + 97T^{2} \)
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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

−10.89567706423197120681701405633, −10.09307224558222568342317572287, −9.368678656154217523830540462538, −8.565910687306719924243848800689, −7.07454466644121467035327596248, −6.40509685473105616021603003070, −4.84651664594467246665156180400, −4.02849380592388524570548907508, −3.15367730429890774651553572651, −1.55250784867597881060993843962, 1.30864786711073282769698328730, 3.22982256778607155970356288366, 4.18753996635165214267693401800, 5.52512359678824907726235184739, 6.31936969058313208499300326232, 7.33729532542365812324902201293, 8.387264978175214957996127455161, 8.658619029853119201835993274611, 9.988001217948260734409184863940, 11.31180263597056331023550613017

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