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Properties

Label 2-490-5.4-c3-0-3
Degree $2$
Conductor $490$
Sign $-0.842 + 0.539i$
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 + 7.11i·3-s − 4·4-s + (6.02 + 9.41i)5-s + 14.2·6-s + 8i·8-s − 23.5·9-s + (18.8 − 12.0i)10-s − 65.2·11-s − 28.4i·12-s + 55.0i·13-s + (−66.9 + 42.8i)15-s + 16·16-s − 116. i·17-s + 47.1i·18-s − 98.8·19-s + ⋯
L(s)  = 1  − 0.707i·2-s + 1.36i·3-s − 0.5·4-s + (0.539 + 0.842i)5-s + 0.967·6-s + 0.353i·8-s − 0.872·9-s + (0.595 − 0.381i)10-s − 1.78·11-s − 0.684i·12-s + 1.17i·13-s + (−1.15 + 0.737i)15-s + 0.250·16-s − 1.66i·17-s + 0.616i·18-s − 1.19·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.842 + 0.539i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.842 + 0.539i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(490\)    =    \(2 \cdot 5 \cdot 7^{2}\)
Sign: $-0.842 + 0.539i$
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: \((2,\ 490,\ (\ :3/2),\ -0.842 + 0.539i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.2755315104\)
\(L(\frac12)\) \(\approx\) \(0.2755315104\)
\(L(\frac{5}{2})\) 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 \( 1 + (-6.02 - 9.41i)T \)
7 \( 1 \)
good3 \( 1 - 7.11iT - 27T^{2} \)
11 \( 1 + 65.2T + 1.33e3T^{2} \)
13 \( 1 - 55.0iT - 2.19e3T^{2} \)
17 \( 1 + 116. iT - 4.91e3T^{2} \)
19 \( 1 + 98.8T + 6.85e3T^{2} \)
23 \( 1 + 87.1iT - 1.21e4T^{2} \)
29 \( 1 - 47.4T + 2.43e4T^{2} \)
31 \( 1 - 165.T + 2.97e4T^{2} \)
37 \( 1 + 107. iT - 5.06e4T^{2} \)
41 \( 1 - 280.T + 6.89e4T^{2} \)
43 \( 1 - 236. iT - 7.95e4T^{2} \)
47 \( 1 + 85.8iT - 1.03e5T^{2} \)
53 \( 1 + 473. iT - 1.48e5T^{2} \)
59 \( 1 + 548.T + 2.05e5T^{2} \)
61 \( 1 + 693.T + 2.26e5T^{2} \)
67 \( 1 + 310. iT - 3.00e5T^{2} \)
71 \( 1 + 331.T + 3.57e5T^{2} \)
73 \( 1 + 277. iT - 3.89e5T^{2} \)
79 \( 1 + 645.T + 4.93e5T^{2} \)
83 \( 1 + 315. iT - 5.71e5T^{2} \)
89 \( 1 + 829.T + 7.04e5T^{2} \)
97 \( 1 + 63.7iT - 9.12e5T^{2} \)
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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

−10.82523371477907413333615233468, −10.28439523028422528292809961932, −9.615441203554089902222488961323, −8.836836639500225047591152491856, −7.54163589064425478389426982742, −6.27687226562226675564435997736, −5.01351263520862981420123645441, −4.41877849844340768883506388183, −3.03032660539611910965632435700, −2.33050484851739087554472832160, 0.083784716168923052720160243936, 1.38586121818835185465814144605, 2.64410688363491458273496925364, 4.52186816192697049549254286958, 5.73708780664072662227654452293, 6.09782636454452426472977820940, 7.45842295513543575287500244740, 8.112771832386855558048560511854, 8.578364004940466762670842602936, 10.05032167884609531187177061167

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