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Properties

Label 2-490-5.4-c3-0-52
Degree $2$
Conductor $490$
Sign $0.559 + 0.828i$
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  + 2i·2-s + 3.06i·3-s − 4·4-s + (9.26 − 6.25i)5-s − 6.13·6-s − 8i·8-s + 17.5·9-s + (12.5 + 18.5i)10-s − 49.5·11-s − 12.2i·12-s − 44.6i·13-s + (19.1 + 28.4i)15-s + 16·16-s − 39.2i·17-s + 35.1i·18-s − 56.3·19-s + ⋯
L(s)  = 1  + 0.707i·2-s + 0.590i·3-s − 0.5·4-s + (0.828 − 0.559i)5-s − 0.417·6-s − 0.353i·8-s + 0.651·9-s + (0.395 + 0.586i)10-s − 1.35·11-s − 0.295i·12-s − 0.951i·13-s + (0.330 + 0.489i)15-s + 0.250·16-s − 0.560i·17-s + 0.460i·18-s − 0.679·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.559 + 0.828i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.559 + 0.828i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(490\)    =    \(2 \cdot 5 \cdot 7^{2}\)
Sign: $0.559 + 0.828i$
Analytic conductor: \(28.9109\)
Root analytic conductor: \(5.37688\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{490} (99, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 490,\ (\ :3/2),\ 0.559 + 0.828i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.176182616\)
\(L(\frac12)\) \(\approx\) \(1.176182616\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 2iT \)
5 \( 1 + (-9.26 + 6.25i)T \)
7 \( 1 \)
good3 \( 1 - 3.06iT - 27T^{2} \)
11 \( 1 + 49.5T + 1.33e3T^{2} \)
13 \( 1 + 44.6iT - 2.19e3T^{2} \)
17 \( 1 + 39.2iT - 4.91e3T^{2} \)
19 \( 1 + 56.3T + 6.85e3T^{2} \)
23 \( 1 + 91.6iT - 1.21e4T^{2} \)
29 \( 1 + 281.T + 2.43e4T^{2} \)
31 \( 1 + 70.5T + 2.97e4T^{2} \)
37 \( 1 + 197. iT - 5.06e4T^{2} \)
41 \( 1 + 399.T + 6.89e4T^{2} \)
43 \( 1 + 203. iT - 7.95e4T^{2} \)
47 \( 1 - 68.4iT - 1.03e5T^{2} \)
53 \( 1 + 617. iT - 1.48e5T^{2} \)
59 \( 1 - 480.T + 2.05e5T^{2} \)
61 \( 1 - 23.4T + 2.26e5T^{2} \)
67 \( 1 + 252. iT - 3.00e5T^{2} \)
71 \( 1 - 835.T + 3.57e5T^{2} \)
73 \( 1 - 257. iT - 3.89e5T^{2} \)
79 \( 1 - 773.T + 4.93e5T^{2} \)
83 \( 1 - 1.34e3iT - 5.71e5T^{2} \)
89 \( 1 - 1.30e3T + 7.04e5T^{2} \)
97 \( 1 + 323. iT - 9.12e5T^{2} \)
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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

−10.22031494071425757376655283197, −9.556354886797651319585422850917, −8.608396049334474387828266112457, −7.74020049322567135498569286737, −6.68991719825113944317655586311, −5.35472727387967652514714946047, −5.12405077339372000158522210393, −3.77469829902083793437526763391, −2.18964709249270238556792279714, −0.34682280804198474835273501395, 1.59640571075530466935154380944, 2.28485509556675883593671207336, 3.64063282743756594806880621775, 4.97892172928341738839788044905, 6.05100758764462011481250173266, 7.05557054574614666795494223802, 7.926555879642875569922839738066, 9.138507980763628860162570553463, 9.978652675281096771766697715287, 10.63903473067417884294847018110

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