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Properties

Label 2-490-245.108-c1-0-11
Degree $2$
Conductor $490$
Sign $0.978 + 0.205i$
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  + (−0.804 + 0.593i)2-s + (0.236 − 1.25i)3-s + (0.294 − 0.955i)4-s + (−2.20 + 0.381i)5-s + (0.552 + 1.14i)6-s + (2.56 + 0.631i)7-s + (0.330 + 0.943i)8-s + (1.28 + 0.504i)9-s + (1.54 − 1.61i)10-s + (−0.352 − 0.898i)11-s + (−1.12 − 0.594i)12-s + (0.328 + 2.91i)13-s + (−2.44 + 1.01i)14-s + (−0.0439 + 2.84i)15-s + (−0.826 − 0.563i)16-s + (−0.267 − 7.15i)17-s + ⋯
L(s)  = 1  + (−0.568 + 0.419i)2-s + (0.136 − 0.722i)3-s + (0.147 − 0.477i)4-s + (−0.985 + 0.170i)5-s + (0.225 + 0.468i)6-s + (0.971 + 0.238i)7-s + (0.116 + 0.333i)8-s + (0.428 + 0.168i)9-s + (0.488 − 0.510i)10-s + (−0.106 − 0.271i)11-s + (−0.324 − 0.171i)12-s + (0.0909 + 0.807i)13-s + (−0.652 + 0.271i)14-s + (−0.0113 + 0.734i)15-s + (−0.206 − 0.140i)16-s + (−0.0649 − 1.73i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.978 + 0.205i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.978 + 0.205i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(490\)    =    \(2 \cdot 5 \cdot 7^{2}\)
Sign: $0.978 + 0.205i$
Analytic conductor: \(3.91266\)
Root analytic conductor: \(1.97804\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{490} (353, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 490,\ (\ :1/2),\ 0.978 + 0.205i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.11003 - 0.115296i\)
\(L(\frac12)\) \(\approx\) \(1.11003 - 0.115296i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.804 - 0.593i)T \)
5 \( 1 + (2.20 - 0.381i)T \)
7 \( 1 + (-2.56 - 0.631i)T \)
good3 \( 1 + (-0.236 + 1.25i)T + (-2.79 - 1.09i)T^{2} \)
11 \( 1 + (0.352 + 0.898i)T + (-8.06 + 7.48i)T^{2} \)
13 \( 1 + (-0.328 - 2.91i)T + (-12.6 + 2.89i)T^{2} \)
17 \( 1 + (0.267 + 7.15i)T + (-16.9 + 1.27i)T^{2} \)
19 \( 1 + (-1.34 - 2.32i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.35 - 0.0879i)T + (22.9 + 1.71i)T^{2} \)
29 \( 1 + (-5.21 - 1.18i)T + (26.1 + 12.5i)T^{2} \)
31 \( 1 + (-4.84 - 2.79i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.26 + 2.40i)T + (-20.8 - 30.5i)T^{2} \)
41 \( 1 + (-4.46 + 9.27i)T + (-25.5 - 32.0i)T^{2} \)
43 \( 1 + (2.59 + 0.908i)T + (33.6 + 26.8i)T^{2} \)
47 \( 1 + (5.97 + 8.09i)T + (-13.8 + 44.9i)T^{2} \)
53 \( 1 + (-1.59 - 3.02i)T + (-29.8 + 43.7i)T^{2} \)
59 \( 1 + (0.0220 - 0.294i)T + (-58.3 - 8.79i)T^{2} \)
61 \( 1 + (-1.27 - 4.13i)T + (-50.4 + 34.3i)T^{2} \)
67 \( 1 + (-1.01 - 3.77i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (-2.94 - 12.9i)T + (-63.9 + 30.8i)T^{2} \)
73 \( 1 + (5.16 - 6.99i)T + (-21.5 - 69.7i)T^{2} \)
79 \( 1 + (-1.04 + 0.600i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (0.294 + 0.0331i)T + (80.9 + 18.4i)T^{2} \)
89 \( 1 + (4.93 - 12.5i)T + (-65.2 - 60.5i)T^{2} \)
97 \( 1 + (9.18 + 9.18i)T + 97iT^{2} \)
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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

−11.06538141637191479916700929098, −9.990118132406320438382545114751, −8.772871536126566828164905137197, −8.183926494394624479754958754263, −7.23535760942428834552444815245, −6.88393524292381902668086903331, −5.29433239090600821401254186378, −4.32260628541654531069795347282, −2.57906964008807453132465784144, −1.07521577718390344447529361618, 1.22135771632944856271008477407, 3.09982339741247421913757225402, 4.20693433153652236612909230903, 4.84103358331459162283510313907, 6.57028346424858342879935034573, 7.909934278835607076492112668192, 8.143185613800091615827406642836, 9.261482082472199688273796019896, 10.27207429344749211473054606470, 10.83199713244416536710507340203

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