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Properties

Label 2-490-35.9-c1-0-9
Degree $2$
Conductor $490$
Sign $0.943 - 0.330i$
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  + (−0.866 − 0.5i)2-s + (−2.59 + 1.5i)3-s + (0.499 + 0.866i)4-s + (2.23 + 0.133i)5-s + 3·6-s − 0.999i·8-s + (3 − 5.19i)9-s + (−1.86 − 1.23i)10-s + (−2.59 − 1.50i)12-s − 2i·13-s + (−6 + 3i)15-s + (−0.5 + 0.866i)16-s + (1.73 − i)17-s + (−5.19 + 3i)18-s + (1 − 1.73i)19-s + (0.999 + 1.99i)20-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (−1.49 + 0.866i)3-s + (0.249 + 0.433i)4-s + (0.998 + 0.0599i)5-s + 1.22·6-s − 0.353i·8-s + (1 − 1.73i)9-s + (−0.590 − 0.389i)10-s + (−0.749 − 0.433i)12-s − 0.554i·13-s + (−1.54 + 0.774i)15-s + (−0.125 + 0.216i)16-s + (0.420 − 0.242i)17-s + (−1.22 + 0.707i)18-s + (0.229 − 0.397i)19-s + (0.223 + 0.447i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.943 - 0.330i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.943 - 0.330i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(490\)    =    \(2 \cdot 5 \cdot 7^{2}\)
Sign: $0.943 - 0.330i$
Analytic conductor: \(3.91266\)
Root analytic conductor: \(1.97804\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{490} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 490,\ (\ :1/2),\ 0.943 - 0.330i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.762029 + 0.129590i\)
\(L(\frac12)\) \(\approx\) \(0.762029 + 0.129590i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 + 0.5i)T \)
5 \( 1 + (-2.23 - 0.133i)T \)
7 \( 1 \)
good3 \( 1 + (2.59 - 1.5i)T + (1.5 - 2.59i)T^{2} \)
11 \( 1 + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 2iT - 13T^{2} \)
17 \( 1 + (-1.73 + i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1 + 1.73i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.866 + 0.5i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - T + 29T^{2} \)
31 \( 1 + (-5 - 8.66i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-6.92 - 4i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 3T + 41T^{2} \)
43 \( 1 + 5iT - 43T^{2} \)
47 \( 1 + (-6.92 - 4i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-5.19 + 3i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (1 + 1.73i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (4.5 - 7.79i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.06 - 3.5i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 6T + 71T^{2} \)
73 \( 1 + (8.66 - 5i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (5 - 8.66i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 9iT - 83T^{2} \)
89 \( 1 + (-3.5 + 6.06i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 97T^{2} \)
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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

−10.73184133664970128746774442560, −10.24925061485807643356133720103, −9.628991357771708281830379649135, −8.673616578565280965258634694396, −7.17895501754613818355807392718, −6.19577552967707319736947817390, −5.43560639884435732202053555320, −4.47060047473403568072882958158, −2.92051508017496846724223945429, −1.00916921836375522427923156611, 0.975520410458260499901207174180, 2.16947919147953350913857254703, 4.63395977760822236284572903696, 5.88428657233783682941557740431, 6.05766709280086773611765814882, 7.12258182110801217283522717325, 7.953842943244093041595531559095, 9.286876046431379508564481736577, 10.09087909831067612946409042335, 10.89640103943661076438208677620

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