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Properties

Label 2-490-245.204-c1-0-27
Degree $2$
Conductor $490$
Sign $-0.723 - 0.690i$
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  + (0.781 − 0.623i)2-s + (−1.36 − 2.82i)3-s + (0.222 − 0.974i)4-s + (−0.747 − 2.10i)5-s + (−2.82 − 1.36i)6-s + (−1.52 + 2.16i)7-s + (−0.433 − 0.900i)8-s + (−4.27 + 5.35i)9-s + (−1.89 − 1.18i)10-s + (−0.773 − 0.969i)11-s + (−3.05 + 0.698i)12-s + (4.30 − 3.43i)13-s + (0.153 + 2.64i)14-s + (−4.94 + 4.98i)15-s + (−0.900 − 0.433i)16-s + (−1.89 + 0.431i)17-s + ⋯
L(s)  = 1  + (0.552 − 0.440i)2-s + (−0.786 − 1.63i)3-s + (0.111 − 0.487i)4-s + (−0.334 − 0.942i)5-s + (−1.15 − 0.555i)6-s + (−0.577 + 0.816i)7-s + (−0.153 − 0.318i)8-s + (−1.42 + 1.78i)9-s + (−0.600 − 0.373i)10-s + (−0.233 − 0.292i)11-s + (−0.883 + 0.201i)12-s + (1.19 − 0.951i)13-s + (0.0409 + 0.705i)14-s + (−1.27 + 1.28i)15-s + (−0.225 − 0.108i)16-s + (−0.458 + 0.104i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.723 - 0.690i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.723 - 0.690i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(490\)    =    \(2 \cdot 5 \cdot 7^{2}\)
Sign: $-0.723 - 0.690i$
Analytic conductor: \(3.91266\)
Root analytic conductor: \(1.97804\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{490} (449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 490,\ (\ :1/2),\ -0.723 - 0.690i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.340983 + 0.851851i\)
\(L(\frac12)\) \(\approx\) \(0.340983 + 0.851851i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.781 + 0.623i)T \)
5 \( 1 + (0.747 + 2.10i)T \)
7 \( 1 + (1.52 - 2.16i)T \)
good3 \( 1 + (1.36 + 2.82i)T + (-1.87 + 2.34i)T^{2} \)
11 \( 1 + (0.773 + 0.969i)T + (-2.44 + 10.7i)T^{2} \)
13 \( 1 + (-4.30 + 3.43i)T + (2.89 - 12.6i)T^{2} \)
17 \( 1 + (1.89 - 0.431i)T + (15.3 - 7.37i)T^{2} \)
19 \( 1 - 2.48T + 19T^{2} \)
23 \( 1 + (3.40 + 0.778i)T + (20.7 + 9.97i)T^{2} \)
29 \( 1 + (0.423 + 1.85i)T + (-26.1 + 12.5i)T^{2} \)
31 \( 1 + 0.154T + 31T^{2} \)
37 \( 1 + (-5.27 + 1.20i)T + (33.3 - 16.0i)T^{2} \)
41 \( 1 + (-6.23 + 3.00i)T + (25.5 - 32.0i)T^{2} \)
43 \( 1 + (0.374 - 0.778i)T + (-26.8 - 33.6i)T^{2} \)
47 \( 1 + (7.75 - 6.18i)T + (10.4 - 45.8i)T^{2} \)
53 \( 1 + (10.8 + 2.47i)T + (47.7 + 22.9i)T^{2} \)
59 \( 1 + (13.2 + 6.37i)T + (36.7 + 46.1i)T^{2} \)
61 \( 1 + (2.42 + 10.6i)T + (-54.9 + 26.4i)T^{2} \)
67 \( 1 + 10.7iT - 67T^{2} \)
71 \( 1 + (2.64 - 11.6i)T + (-63.9 - 30.8i)T^{2} \)
73 \( 1 + (-5.96 - 4.75i)T + (16.2 + 71.1i)T^{2} \)
79 \( 1 - 12.9T + 79T^{2} \)
83 \( 1 + (7.66 + 6.11i)T + (18.4 + 80.9i)T^{2} \)
89 \( 1 + (-7.92 + 9.93i)T + (-19.8 - 86.7i)T^{2} \)
97 \( 1 - 1.61iT - 97T^{2} \)
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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

−11.00082029090964869287970557512, −9.499403209259159486012837345720, −8.348900704754140948258388927196, −7.72720007245274279081554518022, −6.20169184537816053041070523728, −5.95814501885098611266362262755, −4.90910485420995406379833186202, −3.21978252326684058739586305295, −1.81528678160373617555803916692, −0.50631359380687349440454918454, 3.20185707242156532958846724733, 3.96203041317873502923151612420, 4.64464374339646727568858417392, 6.03021301043396083809799846018, 6.53157122807092516503652912427, 7.68775296024571278471035196072, 9.113031194749399384512627604427, 9.908513234992710275293311769613, 10.78116928277668077405355726344, 11.25060087577950950541132897786

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