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Properties

Label 2-490-5.4-c3-0-0
Degree $2$
Conductor $490$
Sign $0.672 + 0.740i$
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 + 10.2i·3-s − 4·4-s + (8.27 − 7.51i)5-s − 20.5·6-s − 8i·8-s − 78.4·9-s + (15.0 + 16.5i)10-s − 30.8·11-s − 41.0i·12-s + 53.3i·13-s + (77.1 + 84.9i)15-s + 16·16-s + 1.41i·17-s − 156. i·18-s − 88.3·19-s + ⋯
L(s)  = 1  + 0.707i·2-s + 1.97i·3-s − 0.5·4-s + (0.740 − 0.672i)5-s − 1.39·6-s − 0.353i·8-s − 2.90·9-s + (0.475 + 0.523i)10-s − 0.846·11-s − 0.987i·12-s + 1.13i·13-s + (1.32 + 1.46i)15-s + 0.250·16-s + 0.0202i·17-s − 2.05i·18-s − 1.06·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.672 + 0.740i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.672 + 0.740i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(490\)    =    \(2 \cdot 5 \cdot 7^{2}\)
Sign: $0.672 + 0.740i$
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: \((2,\ 490,\ (\ :3/2),\ 0.672 + 0.740i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.1228409394\)
\(L(\frac12)\) \(\approx\) \(0.1228409394\)
\(L(\frac{5}{2})\) 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 \( 1 + (-8.27 + 7.51i)T \)
7 \( 1 \)
good3 \( 1 - 10.2iT - 27T^{2} \)
11 \( 1 + 30.8T + 1.33e3T^{2} \)
13 \( 1 - 53.3iT - 2.19e3T^{2} \)
17 \( 1 - 1.41iT - 4.91e3T^{2} \)
19 \( 1 + 88.3T + 6.85e3T^{2} \)
23 \( 1 + 85.0iT - 1.21e4T^{2} \)
29 \( 1 - 49.5T + 2.43e4T^{2} \)
31 \( 1 + 62.7T + 2.97e4T^{2} \)
37 \( 1 + 251. iT - 5.06e4T^{2} \)
41 \( 1 - 197.T + 6.89e4T^{2} \)
43 \( 1 + 93.8iT - 7.95e4T^{2} \)
47 \( 1 - 211. iT - 1.03e5T^{2} \)
53 \( 1 - 388. iT - 1.48e5T^{2} \)
59 \( 1 + 384.T + 2.05e5T^{2} \)
61 \( 1 - 114.T + 2.26e5T^{2} \)
67 \( 1 + 313. iT - 3.00e5T^{2} \)
71 \( 1 + 345.T + 3.57e5T^{2} \)
73 \( 1 + 381. iT - 3.89e5T^{2} \)
79 \( 1 + 957.T + 4.93e5T^{2} \)
83 \( 1 + 135. iT - 5.71e5T^{2} \)
89 \( 1 + 184.T + 7.04e5T^{2} \)
97 \( 1 - 1.25e3iT - 9.12e5T^{2} \)
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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

−10.81828010283574090772103753818, −10.35679117080016808350515607840, −9.264883153439518699798062439856, −9.018914390167975604446188106454, −8.054528944684717195068674560964, −6.34299149789169964114952964328, −5.54942330021214799546734600059, −4.66769544313349366360102754662, −4.10779318069643588814802795378, −2.51435010970293911902235255555, 0.03692437359096064097415323823, 1.40244373207580276390828802192, 2.42448925194111433360605753809, 3.10823897674575053275857219955, 5.35427207213178461565405510616, 6.06648043442377682140163027376, 7.05920621031446571405167842124, 7.914573762273746024630439894345, 8.661471679741324632295313790240, 9.994282188922532678144121428067

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