↑

Properties

Label 2-490-245.239-c1-0-18
Degree $2$
Conductor $490$
Sign $-0.153 + 0.988i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.781 − 0.623i)2-s + (−1.44 + 2.99i)3-s + (0.222 + 0.974i)4-s + (2.01 − 0.970i)5-s + (2.99 − 1.44i)6-s + (−2.57 − 0.622i)7-s + (0.433 − 0.900i)8-s + (−5.01 − 6.29i)9-s + (−2.18 − 0.497i)10-s + (0.511 − 0.641i)11-s + (−3.24 − 0.739i)12-s + (−0.149 − 0.119i)13-s + (1.62 + 2.08i)14-s + (0.00103 + 7.43i)15-s + (−0.900 + 0.433i)16-s + (−6.62 − 1.51i)17-s + ⋯
L(s)  = 1  + (−0.552 − 0.440i)2-s + (−0.832 + 1.72i)3-s + (0.111 + 0.487i)4-s + (0.900 − 0.434i)5-s + (1.22 − 0.588i)6-s + (−0.971 − 0.235i)7-s + (0.153 − 0.318i)8-s + (−1.67 − 2.09i)9-s + (−0.689 − 0.157i)10-s + (0.154 − 0.193i)11-s + (−0.935 − 0.213i)12-s + (−0.0414 − 0.0330i)13-s + (0.433 + 0.558i)14-s + (0.000266 + 1.91i)15-s + (−0.225 + 0.108i)16-s + (−1.60 − 0.366i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.153 + 0.988i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.153 + 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(490\)    =    \(2 \cdot 5 \cdot 7^{2}\)
Sign: $-0.153 + 0.988i$
Analytic conductor: \(3.91266\)
Root analytic conductor: \(1.97804\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{490} (239, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 490,\ (\ :1/2),\ -0.153 + 0.988i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.197983 - 0.231068i\)
\(L(\frac12)\) \(\approx\) \(0.197983 - 0.231068i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.01 + 0.970i)T \)
7 \( 1 + (2.57 + 0.622i)T \)
good3 \( 1 + (1.44 - 2.99i)T + (-1.87 - 2.34i)T^{2} \)
11 \( 1 + (-0.511 + 0.641i)T + (-2.44 - 10.7i)T^{2} \)
13 \( 1 + (0.149 + 0.119i)T + (2.89 + 12.6i)T^{2} \)
17 \( 1 + (6.62 + 1.51i)T + (15.3 + 7.37i)T^{2} \)
19 \( 1 + 5.28T + 19T^{2} \)
23 \( 1 + (-4.89 + 1.11i)T + (20.7 - 9.97i)T^{2} \)
29 \( 1 + (-1.64 + 7.19i)T + (-26.1 - 12.5i)T^{2} \)
31 \( 1 + 1.77T + 31T^{2} \)
37 \( 1 + (1.95 + 0.445i)T + (33.3 + 16.0i)T^{2} \)
41 \( 1 + (3.32 + 1.60i)T + (25.5 + 32.0i)T^{2} \)
43 \( 1 + (3.11 + 6.46i)T + (-26.8 + 33.6i)T^{2} \)
47 \( 1 + (-8.00 - 6.38i)T + (10.4 + 45.8i)T^{2} \)
53 \( 1 + (4.25 - 0.970i)T + (47.7 - 22.9i)T^{2} \)
59 \( 1 + (9.28 - 4.47i)T + (36.7 - 46.1i)T^{2} \)
61 \( 1 + (-0.643 + 2.81i)T + (-54.9 - 26.4i)T^{2} \)
67 \( 1 + 9.37iT - 67T^{2} \)
71 \( 1 + (-1.73 - 7.60i)T + (-63.9 + 30.8i)T^{2} \)
73 \( 1 + (-0.624 + 0.498i)T + (16.2 - 71.1i)T^{2} \)
79 \( 1 + 5.95T + 79T^{2} \)
83 \( 1 + (-1.26 + 1.01i)T + (18.4 - 80.9i)T^{2} \)
89 \( 1 + (2.70 + 3.39i)T + (-19.8 + 86.7i)T^{2} \)
97 \( 1 + 13.4iT - 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.58326165710471568588851765389, −9.932354160059856539577535400625, −9.096641343649009019158357395012, −8.814038412113525020316676843163, −6.67679873640984651041099599503, −6.02532615948745323439432152649, −4.80121747933012329405495490470, −4.00493309254716579627507433932, −2.67540347911185864091711312711, −0.23245364220670812767618884140, 1.61178454452694068621991693871, 2.62883435248821918936825285189, 5.13041950195488810887106779259, 6.15463909936922566933408844383, 6.68289490149193927399077276581, 7.08325357341205584848425331652, 8.476004321414648374939752360211, 9.205331544794573259138563127560, 10.56238028169640504332743542782, 11.01164794250580692057238420364

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