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Properties

Label 2-490-7.4-c1-0-1
Degree $2$
Conductor $490$
Sign $0.386 - 0.922i$
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 + (−0.499 + 0.866i)4-s + (0.5 + 0.866i)5-s + 0.999·8-s + (1.5 + 2.59i)9-s + (0.499 − 0.866i)10-s + (−2 + 3.46i)11-s − 6·13-s + (−0.5 − 0.866i)16-s + (−1 + 1.73i)17-s + (1.5 − 2.59i)18-s − 0.999·20-s + 3.99·22-s + (−0.499 + 0.866i)25-s + (3 + 5.19i)26-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s + (0.223 + 0.387i)5-s + 0.353·8-s + (0.5 + 0.866i)9-s + (0.158 − 0.273i)10-s + (−0.603 + 1.04i)11-s − 1.66·13-s + (−0.125 − 0.216i)16-s + (−0.242 + 0.420i)17-s + (0.353 − 0.612i)18-s − 0.223·20-s + 0.852·22-s + (−0.0999 + 0.173i)25-s + (0.588 + 1.01i)26-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(490\)    =    \(2 \cdot 5 \cdot 7^{2}\)
Sign: $0.386 - 0.922i$
Analytic conductor: \(3.91266\)
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: \((2,\ 490,\ (\ :1/2),\ 0.386 - 0.922i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.742885 + 0.494154i\)
\(L(\frac12)\) \(\approx\) \(0.742885 + 0.494154i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.5 - 2.59i)T^{2} \)
11 \( 1 + (2 - 3.46i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 6T + 13T^{2} \)
17 \( 1 + (1 - 1.73i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
31 \( 1 + (4 - 6.92i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-5 - 8.66i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 2T + 41T^{2} \)
43 \( 1 - 4T + 43T^{2} \)
47 \( 1 + (4 + 6.92i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1 + 1.73i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-4 + 6.92i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-7 - 12.1i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-6 + 10.3i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 16T + 71T^{2} \)
73 \( 1 + (1 - 1.73i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-4 - 6.92i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 8T + 83T^{2} \)
89 \( 1 + (5 + 8.66i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 2T + 97T^{2} \)
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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

−10.91449885548421353299368692065, −10.07705188758104215579701612426, −9.830163187080914398606857518466, −8.451913079789703340567323713260, −7.50779843881752911088938721503, −6.86651061664799124394628266167, −5.16357616770303555663819537530, −4.46419082400885660387656749542, −2.79429567334652041924205310628, −1.91962626774435033059449131011, 0.59109521942451079983750827339, 2.56922720508448546235304923088, 4.20461696965546731226072918316, 5.27449087921738445375888038421, 6.17434434099249059185973251377, 7.23653273850540955255724464992, 7.993270654708154320940015719060, 9.144393881940340659438393943784, 9.624139854098159614713287465549, 10.58964009007528861941458198028

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