Properties

Label 2-490-5.4-c1-0-11
Degree $2$
Conductor $490$
Sign $0.447 - 0.894i$
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 + (2 + i)5-s i·8-s + 3·9-s + (−1 + 2i)10-s + 3·11-s − 5i·13-s + 16-s + 2i·17-s + 3i·18-s − 5·19-s + (−2 − i)20-s + 3i·22-s + 7i·23-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.5·4-s + (0.894 + 0.447i)5-s − 0.353i·8-s + 9-s + (−0.316 + 0.632i)10-s + 0.904·11-s − 1.38i·13-s + 0.250·16-s + 0.485i·17-s + 0.707i·18-s − 1.14·19-s + (−0.447 − 0.223i)20-s + 0.639i·22-s + 1.45i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(490\)    =    \(2 \cdot 5 \cdot 7^{2}\)
Sign: $0.447 - 0.894i$
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: \((2,\ 490,\ (\ :1/2),\ 0.447 - 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.44834 + 0.895124i\)
\(L(\frac12)\) \(\approx\) \(1.44834 + 0.895124i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2 - i)T \)
7 \( 1 \)
good3 \( 1 - 3T^{2} \)
11 \( 1 - 3T + 11T^{2} \)
13 \( 1 + 5iT - 13T^{2} \)
17 \( 1 - 2iT - 17T^{2} \)
19 \( 1 + 5T + 19T^{2} \)
23 \( 1 - 7iT - 23T^{2} \)
29 \( 1 - 4T + 29T^{2} \)
31 \( 1 - 2T + 31T^{2} \)
37 \( 1 - iT - 37T^{2} \)
41 \( 1 + 3T + 41T^{2} \)
43 \( 1 + 2iT - 43T^{2} \)
47 \( 1 - 7iT - 47T^{2} \)
53 \( 1 + 9iT - 53T^{2} \)
59 \( 1 + 4T + 59T^{2} \)
61 \( 1 + 6T + 61T^{2} \)
67 \( 1 - 2iT - 67T^{2} \)
71 \( 1 + 6T + 71T^{2} \)
73 \( 1 + 16iT - 73T^{2} \)
79 \( 1 + 14T + 79T^{2} \)
83 \( 1 + 6iT - 83T^{2} \)
89 \( 1 - 2T + 89T^{2} \)
97 \( 1 - 12iT - 97T^{2} \)
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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

−10.80507070742835426501614440460, −10.11853796096329643550247952201, −9.396151272125967098321882505444, −8.345356990409241166031175342836, −7.33690888757180750146905209485, −6.46620140662974376235907547777, −5.74452128344921372796566132051, −4.55356671093030157894897454841, −3.30463564635477408817207360554, −1.55411791066255474757878657474, 1.33983583697781415895254106116, 2.38448690973841340693879134936, 4.18123539548456914070844542157, 4.69355457239090258952234151741, 6.25517366307462395317932108514, 6.91510649035026153117224646481, 8.544915215203032126277274070988, 9.145893820676618331685318581887, 9.955802034071169369266024745726, 10.66035997935490810962620368713

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