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Properties

Label 2-490-5.4-c1-0-6
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

\(L(1)\) \(\approx\) \(1.09762 + 0.678370i\)
\(L(\frac12)\) \(\approx\) \(1.09762 + 0.678370i\)
\(L(\frac{3}{2})\) 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

−11.48947733962775198678491193185, −9.858366224944203133113634331727, −9.282546635573790288855844113001, −8.372242676560961015373059379618, −7.18106508689761536870910180359, −6.95227880076570226611336555267, −5.41580486973594427293227040744, −4.38608647121429650792201448110, −3.66343780854420297138124817502, −1.32450061326598305781130059527, 1.01170395624524908976963510299, 2.84863856857504638710452271731, 3.84495003997242296064139441080, 4.73102116268163805107121819294, 6.19760980646821566432054851269, 7.31931761305055367988344293878, 8.088437259162251802541409190148, 9.137085089351067203624728961973, 10.23195487658469570204656855462, 10.69100982535085619094236129805

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