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Properties

Label 2-490-35.9-c1-0-13
Degree $2$
Conductor $490$
Sign $0.291 + 0.956i$
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 + (−2.12 + 1.22i)3-s + (0.499 + 0.866i)4-s + (−1.81 − 1.30i)5-s − 2.44·6-s + 0.999i·8-s + (1.49 − 2.59i)9-s + (−0.917 − 2.03i)10-s + (−2.44 − 4.24i)11-s + (−2.12 − 1.22i)12-s + 0.449i·13-s + (5.44 + 0.550i)15-s + (−0.5 + 0.866i)16-s + (1.73 − i)17-s + (2.59 − 1.49i)18-s + (3.22 − 5.58i)19-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (−1.22 + 0.707i)3-s + (0.249 + 0.433i)4-s + (−0.811 − 0.584i)5-s − 0.999·6-s + 0.353i·8-s + (0.499 − 0.866i)9-s + (−0.290 − 0.644i)10-s + (−0.738 − 1.27i)11-s + (−0.612 − 0.353i)12-s + 0.124i·13-s + (1.40 + 0.142i)15-s + (−0.125 + 0.216i)16-s + (0.420 − 0.242i)17-s + (0.612 − 0.353i)18-s + (0.739 − 1.28i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.291 + 0.956i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.291 + 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(490\)    =    \(2 \cdot 5 \cdot 7^{2}\)
Sign: $0.291 + 0.956i$
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: \((2,\ 490,\ (\ :1/2),\ 0.291 + 0.956i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.473914 - 0.350875i\)
\(L(\frac12)\) \(\approx\) \(0.473914 - 0.350875i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.81 + 1.30i)T \)
7 \( 1 \)
good3 \( 1 + (2.12 - 1.22i)T + (1.5 - 2.59i)T^{2} \)
11 \( 1 + (2.44 + 4.24i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 0.449iT - 13T^{2} \)
17 \( 1 + (-1.73 + i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.22 + 5.58i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (5.97 + 3.44i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 2.89T + 29T^{2} \)
31 \( 1 + (-0.449 - 0.778i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.73 + i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 10.8T + 41T^{2} \)
43 \( 1 + 8.89iT - 43T^{2} \)
47 \( 1 + (-0.778 - 0.449i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-0.953 + 0.550i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (3.22 + 5.58i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (4.22 - 7.31i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.92 - 4i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 10.8T + 71T^{2} \)
73 \( 1 + (-5.97 + 3.44i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (1.44 - 2.51i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 2.44iT - 83T^{2} \)
89 \( 1 + (5 - 8.66i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 3.79iT - 97T^{2} \)
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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

−10.97315353091240197490060331047, −10.21904565818612086136277941282, −8.852933994688059122786618358635, −8.028585870346477484516068397661, −6.89333252752517115868512813171, −5.75105272046740568401500529304, −5.13434265916063874422414056329, −4.31307209826674462961720071112, −3.15652464758420828856568594787, −0.33848482946699426048840121122, 1.66773073990727150558162917393, 3.28251810114047537202158709157, 4.53463212752621269073172856051, 5.54890485282969777790916063294, 6.39991728054786201757716633187, 7.37478505009261801399474043398, 7.955256135143386740446389852970, 9.943754536806956013846474418078, 10.42228268873716670256285520725, 11.45258746322219228150733703815

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