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Properties

Label 2-490-5.4-c1-0-16
Degree $2$
Conductor $490$
Sign $-0.948 + 0.316i$
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  − i·2-s − 4-s + (−0.707 − 2.12i)5-s + i·8-s + 3·9-s + (−2.12 + 0.707i)10-s − 4·11-s − 4.24i·13-s + 16-s − 4.24i·17-s − 3i·18-s − 5.65·19-s + (0.707 + 2.12i)20-s + 4i·22-s + (−3.99 + 3i)25-s − 4.24·26-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.5·4-s + (−0.316 − 0.948i)5-s + 0.353i·8-s + 9-s + (−0.670 + 0.223i)10-s − 1.20·11-s − 1.17i·13-s + 0.250·16-s − 1.02i·17-s − 0.707i·18-s − 1.29·19-s + (0.158 + 0.474i)20-s + 0.852i·22-s + (−0.799 + 0.600i)25-s − 0.832·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.948 + 0.316i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.948 + 0.316i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(490\)    =    \(2 \cdot 5 \cdot 7^{2}\)
Sign: $-0.948 + 0.316i$
Analytic conductor: \(3.91266\)
Root analytic conductor: \(1.97804\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{490} (99, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 490,\ (\ :1/2),\ -0.948 + 0.316i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.152546 - 0.940036i\)
\(L(\frac12)\) \(\approx\) \(0.152546 - 0.940036i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
5 \( 1 + (0.707 + 2.12i)T \)
7 \( 1 \)
good3 \( 1 - 3T^{2} \)
11 \( 1 + 4T + 11T^{2} \)
13 \( 1 + 4.24iT - 13T^{2} \)
17 \( 1 + 4.24iT - 17T^{2} \)
19 \( 1 + 5.65T + 19T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 - 4T + 29T^{2} \)
31 \( 1 + 5.65T + 31T^{2} \)
37 \( 1 - 6iT - 37T^{2} \)
41 \( 1 + 1.41T + 41T^{2} \)
43 \( 1 + 12iT - 43T^{2} \)
47 \( 1 - 47T^{2} \)
53 \( 1 + 12iT - 53T^{2} \)
59 \( 1 - 11.3T + 59T^{2} \)
61 \( 1 - 7.07T + 61T^{2} \)
67 \( 1 - 12iT - 67T^{2} \)
71 \( 1 - 8T + 71T^{2} \)
73 \( 1 - 4.24iT - 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + 16.9iT - 83T^{2} \)
89 \( 1 - 4.24T + 89T^{2} \)
97 \( 1 - 4.24iT - 97T^{2} \)
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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

−10.39405382622778599262283870541, −9.973369818705712851570709956253, −8.744985478837578697986879679787, −8.081689265185374209991837963058, −7.08586640332681633484634916990, −5.42167910791288474750534438104, −4.79002352062115259337577909829, −3.64396385340301842744383990390, −2.22551720897842378576468987380, −0.56814434388515239751150233384, 2.16725468150336597504109165035, 3.79306753552570853388467068280, 4.63631292481258141598241705645, 6.06342735657958662737718560322, 6.81397173947513738962393304972, 7.60765203584006582709366414271, 8.436908528800116244632026511627, 9.616480939241198437459596924242, 10.49852243863850044048183583117, 11.07453987657362535529023391091

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