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Properties

Label 2-490-35.33-c1-0-2
Degree $2$
Conductor $490$
Sign $0.294 - 0.955i$
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  + (−0.258 − 0.965i)2-s + (−1.78 − 0.478i)3-s + (−0.866 + 0.499i)4-s + (1.85 + 1.25i)5-s + 1.84i·6-s + (0.707 + 0.707i)8-s + (0.358 + 0.207i)9-s + (0.730 − 2.11i)10-s + (−1.41 − 2.44i)11-s + (1.78 − 0.478i)12-s + (−4.23 + 4.23i)13-s + (−2.70 − 3.12i)15-s + (0.500 − 0.866i)16-s + (−1.35 + 5.04i)17-s + (0.107 − 0.400i)18-s + (−0.699 + 1.21i)19-s + ⋯
L(s)  = 1  + (−0.183 − 0.683i)2-s + (−1.03 − 0.276i)3-s + (−0.433 + 0.249i)4-s + (0.828 + 0.560i)5-s + 0.754i·6-s + (0.249 + 0.249i)8-s + (0.119 + 0.0690i)9-s + (0.230 − 0.668i)10-s + (−0.426 − 0.738i)11-s + (0.515 − 0.138i)12-s + (−1.17 + 1.17i)13-s + (−0.698 − 0.805i)15-s + (0.125 − 0.216i)16-s + (−0.328 + 1.22i)17-s + (0.0252 − 0.0943i)18-s + (−0.160 + 0.278i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.294 - 0.955i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.294 - 0.955i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(490\)    =    \(2 \cdot 5 \cdot 7^{2}\)
Sign: $0.294 - 0.955i$
Analytic conductor: \(3.91266\)
Root analytic conductor: \(1.97804\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{490} (313, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 490,\ (\ :1/2),\ 0.294 - 0.955i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.401100 + 0.295979i\)
\(L(\frac12)\) \(\approx\) \(0.401100 + 0.295979i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.258 + 0.965i)T \)
5 \( 1 + (-1.85 - 1.25i)T \)
7 \( 1 \)
good3 \( 1 + (1.78 + 0.478i)T + (2.59 + 1.5i)T^{2} \)
11 \( 1 + (1.41 + 2.44i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (4.23 - 4.23i)T - 13iT^{2} \)
17 \( 1 + (1.35 - 5.04i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (0.699 - 1.21i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.565 - 0.151i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 - 0.828iT - 29T^{2} \)
31 \( 1 + (1.32 - 0.765i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.946 - 3.53i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + 3.69iT - 41T^{2} \)
43 \( 1 + (-4 - 4i)T + 43iT^{2} \)
47 \( 1 + (1.47 - 0.396i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (3.01 - 11.2i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (-4.61 - 8.00i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (5.57 + 3.21i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (14.3 + 3.83i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + 0.585T + 71T^{2} \)
73 \( 1 + (5.66 + 1.51i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-4.39 - 2.53i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-5.31 + 5.31i)T - 83iT^{2} \)
89 \( 1 + (5.67 - 9.82i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (4.59 + 4.59i)T + 97iT^{2} \)
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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

−10.98276214060580278421159454646, −10.56790849480383324381738237932, −9.578717193220557038839496926810, −8.698338258100121577630501810442, −7.36132551443363970384109604587, −6.34161242553759277256305472234, −5.66355307990680226244609124424, −4.46933726823069702054881868889, −2.92686036831328746979224276916, −1.67419371226985228539622936681, 0.35167743132234177084915877518, 2.47980402995328084817161345397, 4.72310935228166451477857901601, 5.13582069528308919289160320820, 5.93436643202332658154570885991, 7.02443354950392996765911153223, 7.944966348458595771863920730695, 9.141390077012951202366269565694, 9.932207699666483542332490570681, 10.50305466812339453078081669934

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