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Properties

Label 2-490-245.103-c1-0-24
Degree $2$
Conductor $490$
Sign $0.0947 + 0.995i$
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  + (0.593 − 0.804i)2-s + (1.94 − 0.367i)3-s + (−0.294 − 0.955i)4-s + (0.178 − 2.22i)5-s + (0.856 − 1.77i)6-s + (2.63 + 0.244i)7-s + (−0.943 − 0.330i)8-s + (0.838 − 0.329i)9-s + (−1.68 − 1.46i)10-s + (−0.564 + 1.43i)11-s + (−0.922 − 1.74i)12-s + (−2.14 − 0.242i)13-s + (1.76 − 1.97i)14-s + (−0.472 − 4.39i)15-s + (−0.826 + 0.563i)16-s + (2.71 + 0.101i)17-s + ⋯
L(s)  = 1  + (0.419 − 0.568i)2-s + (1.12 − 0.211i)3-s + (−0.147 − 0.477i)4-s + (0.0798 − 0.996i)5-s + (0.349 − 0.726i)6-s + (0.995 + 0.0924i)7-s + (−0.333 − 0.116i)8-s + (0.279 − 0.109i)9-s + (−0.533 − 0.463i)10-s + (−0.170 + 0.433i)11-s + (−0.266 − 0.504i)12-s + (−0.595 − 0.0671i)13-s + (0.470 − 0.527i)14-s + (−0.121 − 1.13i)15-s + (−0.206 + 0.140i)16-s + (0.658 + 0.0246i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0947 + 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0947 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(490\)    =    \(2 \cdot 5 \cdot 7^{2}\)
Sign: $0.0947 + 0.995i$
Analytic conductor: \(3.91266\)
Root analytic conductor: \(1.97804\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{490} (103, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 490,\ (\ :1/2),\ 0.0947 + 0.995i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.88935 - 1.71800i\)
\(L(\frac12)\) \(\approx\) \(1.88935 - 1.71800i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.593 + 0.804i)T \)
5 \( 1 + (-0.178 + 2.22i)T \)
7 \( 1 + (-2.63 - 0.244i)T \)
good3 \( 1 + (-1.94 + 0.367i)T + (2.79 - 1.09i)T^{2} \)
11 \( 1 + (0.564 - 1.43i)T + (-8.06 - 7.48i)T^{2} \)
13 \( 1 + (2.14 + 0.242i)T + (12.6 + 2.89i)T^{2} \)
17 \( 1 + (-2.71 - 0.101i)T + (16.9 + 1.27i)T^{2} \)
19 \( 1 + (-0.197 + 0.341i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.204 - 5.45i)T + (-22.9 + 1.71i)T^{2} \)
29 \( 1 + (4.56 - 1.04i)T + (26.1 - 12.5i)T^{2} \)
31 \( 1 + (-1.90 + 1.09i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-7.58 + 4.00i)T + (20.8 - 30.5i)T^{2} \)
41 \( 1 + (3.78 + 7.86i)T + (-25.5 + 32.0i)T^{2} \)
43 \( 1 + (-3.37 - 9.65i)T + (-33.6 + 26.8i)T^{2} \)
47 \( 1 + (-2.60 - 1.92i)T + (13.8 + 44.9i)T^{2} \)
53 \( 1 + (7.26 + 3.84i)T + (29.8 + 43.7i)T^{2} \)
59 \( 1 + (-0.473 - 6.32i)T + (-58.3 + 8.79i)T^{2} \)
61 \( 1 + (2.00 - 6.50i)T + (-50.4 - 34.3i)T^{2} \)
67 \( 1 + (3.63 + 0.973i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (-3.01 + 13.2i)T + (-63.9 - 30.8i)T^{2} \)
73 \( 1 + (-5.81 + 4.29i)T + (21.5 - 69.7i)T^{2} \)
79 \( 1 + (-6.15 - 3.55i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-0.306 - 2.72i)T + (-80.9 + 18.4i)T^{2} \)
89 \( 1 + (-4.17 - 10.6i)T + (-65.2 + 60.5i)T^{2} \)
97 \( 1 + (-3.70 - 3.70i)T + 97iT^{2} \)
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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

−10.86695955077205464973062148533, −9.568740726184770719546172504995, −9.157225252962056994645950987935, −7.993666745142591622487948044249, −7.58170201435160192765593257938, −5.67963461381941715859192295552, −4.90016162346643163120521461109, −3.83825976340137272423562249732, −2.48147737706957552030679906572, −1.48766700709466738808296467652, 2.33181743889257666373355274788, 3.25301130443593854729717718461, 4.33958450136967216571351276454, 5.55032428775525278455546757873, 6.66998043142039593508442324231, 7.77479020129525821979831561038, 8.140808633192869106448693110700, 9.243171154297786868625887300121, 10.19731874947394116783079492216, 11.17790302097940880786802065117

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