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Properties

Label 2-490-7.2-c3-0-6
Degree $2$
Conductor $490$
Sign $-0.386 - 0.922i$
Analytic cond. $28.9109$
Root an. cond. $5.37688$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1 − 1.73i)2-s + (1.5 + 2.59i)3-s + (−1.99 − 3.46i)4-s + (−2.5 + 4.33i)5-s + 6·6-s − 7.99·8-s + (9 − 15.5i)9-s + (5 + 8.66i)10-s + (8.5 + 14.7i)11-s + (6.00 − 10.3i)12-s − 81·13-s − 15.0·15-s + (−8 + 13.8i)16-s + (45.5 + 78.8i)17-s + (−18 − 31.1i)18-s + (−51 + 88.3i)19-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 + (0.232 + 0.403i)11-s + (0.144 − 0.249i)12-s − 1.72·13-s − 0.258·15-s + (−0.125 + 0.216i)16-s + (0.649 + 1.12i)17-s + (−0.235 − 0.408i)18-s + (−0.615 + 1.06i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.386 - 0.922i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(490\)    =    \(2 \cdot 5 \cdot 7^{2}\)
Sign: $-0.386 - 0.922i$
Analytic conductor: \(28.9109\)
Root analytic conductor: \(5.37688\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{490} (471, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 490,\ (\ :3/2),\ -0.386 - 0.922i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.002880658\)
\(L(\frac12)\) \(\approx\) \(1.002880658\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1 + 1.73i)T \)
5 \( 1 + (2.5 - 4.33i)T \)
7 \( 1 \)
good3 \( 1 + (-1.5 - 2.59i)T + (-13.5 + 23.3i)T^{2} \)
11 \( 1 + (-8.5 - 14.7i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + 81T + 2.19e3T^{2} \)
17 \( 1 + (-45.5 - 78.8i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (51 - 88.3i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-45 + 77.9i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + 129T + 2.43e4T^{2} \)
31 \( 1 + (58 + 100. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (157 - 271. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + 124T + 6.89e4T^{2} \)
43 \( 1 + 434T + 7.95e4T^{2} \)
47 \( 1 + (248.5 - 430. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (-292 - 505. i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (-166 - 287. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (110 - 190. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (192 + 332. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 664T + 3.57e5T^{2} \)
73 \( 1 + (115 + 199. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (180.5 - 312. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 - 1.17e3T + 5.71e5T^{2} \)
89 \( 1 + (20 - 34.6i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + 175T + 9.12e5T^{2} \)
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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

−10.57008077507323082531613537551, −10.09763049182849821061936742125, −9.379049872101267325123945917934, −8.236397924584690690118901958643, −7.14452600246217062658953551468, −6.10780251559037002648656274849, −4.79712136615398120286490716957, −3.96462540800582875329562099558, −3.00826073633262746989049859498, −1.67905153712234932726601862466, 0.25284674047600444708128237707, 2.07461690951247887983695099257, 3.35778008061874873853911664770, 4.85088696188358942287210006021, 5.28383253281939190261231757165, 7.02088306134278847687574229716, 7.23690921335504237567878646511, 8.292144243457909312595558124102, 9.176855228990429572957950978137, 10.10655963713706728285976052855

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