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Properties

Label 2-490-245.103-c1-0-7
Degree $2$
Conductor $490$
Sign $0.619 - 0.784i$
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 + (−1.35 + 0.256i)3-s + (−0.294 − 0.955i)4-s + (1.70 + 1.44i)5-s + (−0.598 + 1.24i)6-s + (1.54 + 2.15i)7-s + (−0.943 − 0.330i)8-s + (−1.02 + 0.401i)9-s + (2.17 − 0.518i)10-s + (−1.42 + 3.61i)11-s + (0.644 + 1.21i)12-s + (−2.17 − 0.244i)13-s + (2.64 + 0.0369i)14-s + (−2.68 − 1.51i)15-s + (−0.826 + 0.563i)16-s + (−1.18 − 0.0442i)17-s + ⋯
L(s)  = 1  + (0.419 − 0.568i)2-s + (−0.781 + 0.147i)3-s + (−0.147 − 0.477i)4-s + (0.764 + 0.645i)5-s + (−0.244 + 0.507i)6-s + (0.582 + 0.812i)7-s + (−0.333 − 0.116i)8-s + (−0.341 + 0.133i)9-s + (0.687 − 0.163i)10-s + (−0.428 + 1.09i)11-s + (0.185 + 0.351i)12-s + (−0.602 − 0.0678i)13-s + (0.707 + 0.00987i)14-s + (−0.692 − 0.391i)15-s + (−0.206 + 0.140i)16-s + (−0.286 − 0.0107i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.619 - 0.784i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.619 - 0.784i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(490\)    =    \(2 \cdot 5 \cdot 7^{2}\)
Sign: $0.619 - 0.784i$
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: \((2,\ 490,\ (\ :1/2),\ 0.619 - 0.784i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.18569 + 0.574620i\)
\(L(\frac12)\) \(\approx\) \(1.18569 + 0.574620i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.70 - 1.44i)T \)
7 \( 1 + (-1.54 - 2.15i)T \)
good3 \( 1 + (1.35 - 0.256i)T + (2.79 - 1.09i)T^{2} \)
11 \( 1 + (1.42 - 3.61i)T + (-8.06 - 7.48i)T^{2} \)
13 \( 1 + (2.17 + 0.244i)T + (12.6 + 2.89i)T^{2} \)
17 \( 1 + (1.18 + 0.0442i)T + (16.9 + 1.27i)T^{2} \)
19 \( 1 + (0.548 - 0.949i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.317 - 8.49i)T + (-22.9 + 1.71i)T^{2} \)
29 \( 1 + (-10.0 + 2.29i)T + (26.1 - 12.5i)T^{2} \)
31 \( 1 + (-5.29 + 3.05i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (3.04 - 1.60i)T + (20.8 - 30.5i)T^{2} \)
41 \( 1 + (3.48 + 7.24i)T + (-25.5 + 32.0i)T^{2} \)
43 \( 1 + (0.766 + 2.18i)T + (-33.6 + 26.8i)T^{2} \)
47 \( 1 + (-1.57 - 1.16i)T + (13.8 + 44.9i)T^{2} \)
53 \( 1 + (1.34 + 0.713i)T + (29.8 + 43.7i)T^{2} \)
59 \( 1 + (-0.689 - 9.20i)T + (-58.3 + 8.79i)T^{2} \)
61 \( 1 + (-2.81 + 9.12i)T + (-50.4 - 34.3i)T^{2} \)
67 \( 1 + (-1.05 - 0.282i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (-1.45 + 6.39i)T + (-63.9 - 30.8i)T^{2} \)
73 \( 1 + (9.21 - 6.80i)T + (21.5 - 69.7i)T^{2} \)
79 \( 1 + (-2.30 - 1.32i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-0.712 - 6.32i)T + (-80.9 + 18.4i)T^{2} \)
89 \( 1 + (4.09 + 10.4i)T + (-65.2 + 60.5i)T^{2} \)
97 \( 1 + (11.6 + 11.6i)T + 97iT^{2} \)
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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

−11.21302683721950357239320181996, −10.24194718676277614948675488916, −9.780600780837772643383784230640, −8.543392106683962137506988589441, −7.24703260907681633818842079615, −6.11446591344054460953047328336, −5.35454749748026050689434856056, −4.65450606022506138669328707789, −2.85372722022488669623165964346, −1.94619679697146238293614315892, 0.76833594147246666977664271774, 2.81509835779509328264779564270, 4.60777894427191088148797792752, 5.08842904655310007886003301105, 6.21865000516029243039884906331, 6.77352035494260421280907628574, 8.269885475851349101023635696125, 8.669944715737736820382380112019, 10.16081318083162045264766534897, 10.83126213572568292239767243130

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