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Properties

Label 2-490-245.103-c1-0-27
Degree $2$
Conductor $490$
Sign $-0.814 - 0.580i$
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 + (−0.561 + 0.106i)3-s + (−0.294 − 0.955i)4-s + (−2.22 + 0.211i)5-s + (−0.248 + 0.515i)6-s + (1.35 − 2.27i)7-s + (−0.943 − 0.330i)8-s + (−2.48 + 0.976i)9-s + (−1.15 + 1.91i)10-s + (−1.52 + 3.88i)11-s + (0.267 + 0.505i)12-s + (−3.96 − 0.446i)13-s + (−1.02 − 2.43i)14-s + (1.22 − 0.355i)15-s + (−0.826 + 0.563i)16-s + (−3.04 − 0.113i)17-s + ⋯
L(s)  = 1  + (0.419 − 0.568i)2-s + (−0.324 + 0.0613i)3-s + (−0.147 − 0.477i)4-s + (−0.995 + 0.0944i)5-s + (−0.101 + 0.210i)6-s + (0.511 − 0.859i)7-s + (−0.333 − 0.116i)8-s + (−0.829 + 0.325i)9-s + (−0.364 + 0.606i)10-s + (−0.459 + 1.17i)11-s + (0.0771 + 0.145i)12-s + (−1.10 − 0.123i)13-s + (−0.274 − 0.651i)14-s + (0.317 − 0.0917i)15-s + (−0.206 + 0.140i)16-s + (−0.738 − 0.0276i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.814 - 0.580i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.814 - 0.580i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(490\)    =    \(2 \cdot 5 \cdot 7^{2}\)
Sign: $-0.814 - 0.580i$
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: \((2,\ 490,\ (\ :1/2),\ -0.814 - 0.580i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0369057 + 0.115442i\)
\(L(\frac12)\) \(\approx\) \(0.0369057 + 0.115442i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.22 - 0.211i)T \)
7 \( 1 + (-1.35 + 2.27i)T \)
good3 \( 1 + (0.561 - 0.106i)T + (2.79 - 1.09i)T^{2} \)
11 \( 1 + (1.52 - 3.88i)T + (-8.06 - 7.48i)T^{2} \)
13 \( 1 + (3.96 + 0.446i)T + (12.6 + 2.89i)T^{2} \)
17 \( 1 + (3.04 + 0.113i)T + (16.9 + 1.27i)T^{2} \)
19 \( 1 + (-1.51 + 2.62i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.0241 - 0.646i)T + (-22.9 + 1.71i)T^{2} \)
29 \( 1 + (5.02 - 1.14i)T + (26.1 - 12.5i)T^{2} \)
31 \( 1 + (6.81 - 3.93i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.13 - 0.600i)T + (20.8 - 30.5i)T^{2} \)
41 \( 1 + (0.746 + 1.54i)T + (-25.5 + 32.0i)T^{2} \)
43 \( 1 + (0.282 + 0.808i)T + (-33.6 + 26.8i)T^{2} \)
47 \( 1 + (10.4 + 7.71i)T + (13.8 + 44.9i)T^{2} \)
53 \( 1 + (-9.78 - 5.17i)T + (29.8 + 43.7i)T^{2} \)
59 \( 1 + (-0.348 - 4.65i)T + (-58.3 + 8.79i)T^{2} \)
61 \( 1 + (-2.72 + 8.82i)T + (-50.4 - 34.3i)T^{2} \)
67 \( 1 + (-10.3 - 2.76i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (-1.57 + 6.90i)T + (-63.9 - 30.8i)T^{2} \)
73 \( 1 + (-9.22 + 6.80i)T + (21.5 - 69.7i)T^{2} \)
79 \( 1 + (8.19 + 4.73i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (0.369 + 3.27i)T + (-80.9 + 18.4i)T^{2} \)
89 \( 1 + (3.67 + 9.36i)T + (-65.2 + 60.5i)T^{2} \)
97 \( 1 + (6.70 + 6.70i)T + 97iT^{2} \)
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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

−10.74711921032274203859984655628, −9.837089058473821562735321277966, −8.632577666117014300887227287610, −7.50285274523805939053574033767, −6.96884547867915588323320418207, −5.17036232276124021785128364197, −4.73192629150294978258813695691, −3.55458656863953552084371479312, −2.19163298117478254686756857697, −0.06192996177958691074462117943, 2.66312245085505898259045489425, 3.83937787367212395025313982291, 5.15007697311823923649212697621, 5.70547064688588682679650208302, 6.88185754827681596020932023534, 7.986269319091144341764268712043, 8.482609383699985896214754407043, 9.456498487243977403667385686815, 11.20341346249195647938940907698, 11.37977269623461575373087159254

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