Properties

Label 2-490-245.103-c1-0-5
Degree $2$
Conductor $490$
Sign $-0.988 - 0.153i$
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 + (−1.03 + 0.194i)3-s + (−0.294 − 0.955i)4-s + (0.328 + 2.21i)5-s + (0.454 − 0.944i)6-s + (1.16 + 2.37i)7-s + (0.943 + 0.330i)8-s + (−1.76 + 0.694i)9-s + (−1.97 − 1.04i)10-s + (0.564 − 1.43i)11-s + (0.489 + 0.927i)12-s + (2.46 + 0.278i)13-s + (−2.60 − 0.477i)14-s + (−0.769 − 2.21i)15-s + (−0.826 + 0.563i)16-s + (−0.883 − 0.0330i)17-s + ⋯
L(s)  = 1  + (−0.419 + 0.568i)2-s + (−0.594 + 0.112i)3-s + (−0.147 − 0.477i)4-s + (0.146 + 0.989i)5-s + (0.185 − 0.385i)6-s + (0.438 + 0.898i)7-s + (0.333 + 0.116i)8-s + (−0.589 + 0.231i)9-s + (−0.624 − 0.331i)10-s + (0.170 − 0.433i)11-s + (0.141 + 0.267i)12-s + (0.684 + 0.0771i)13-s + (−0.695 − 0.127i)14-s + (−0.198 − 0.571i)15-s + (−0.206 + 0.140i)16-s + (−0.214 − 0.00802i)17-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(490\)    =    \(2 \cdot 5 \cdot 7^{2}\)
Sign: $-0.988 - 0.153i$
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: \((2,\ 490,\ (\ :1/2),\ -0.988 - 0.153i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0520239 + 0.672058i\)
\(L(\frac12)\) \(\approx\) \(0.0520239 + 0.672058i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.328 - 2.21i)T \)
7 \( 1 + (-1.16 - 2.37i)T \)
good3 \( 1 + (1.03 - 0.194i)T + (2.79 - 1.09i)T^{2} \)
11 \( 1 + (-0.564 + 1.43i)T + (-8.06 - 7.48i)T^{2} \)
13 \( 1 + (-2.46 - 0.278i)T + (12.6 + 2.89i)T^{2} \)
17 \( 1 + (0.883 + 0.0330i)T + (16.9 + 1.27i)T^{2} \)
19 \( 1 + (1.61 - 2.79i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.0273 - 0.731i)T + (-22.9 + 1.71i)T^{2} \)
29 \( 1 + (10.4 - 2.37i)T + (26.1 - 12.5i)T^{2} \)
31 \( 1 + (1.07 - 0.619i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (3.46 - 1.83i)T + (20.8 - 30.5i)T^{2} \)
41 \( 1 + (1.18 + 2.46i)T + (-25.5 + 32.0i)T^{2} \)
43 \( 1 + (-1.69 - 4.83i)T + (-33.6 + 26.8i)T^{2} \)
47 \( 1 + (3.68 + 2.72i)T + (13.8 + 44.9i)T^{2} \)
53 \( 1 + (0.847 + 0.447i)T + (29.8 + 43.7i)T^{2} \)
59 \( 1 + (0.212 + 2.84i)T + (-58.3 + 8.79i)T^{2} \)
61 \( 1 + (-0.583 + 1.89i)T + (-50.4 - 34.3i)T^{2} \)
67 \( 1 + (-8.81 - 2.36i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (0.892 - 3.91i)T + (-63.9 - 30.8i)T^{2} \)
73 \( 1 + (-5.56 + 4.10i)T + (21.5 - 69.7i)T^{2} \)
79 \( 1 + (-9.38 - 5.41i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (0.469 + 4.16i)T + (-80.9 + 18.4i)T^{2} \)
89 \( 1 + (-0.312 - 0.797i)T + (-65.2 + 60.5i)T^{2} \)
97 \( 1 + (-10.5 - 10.5i)T + 97iT^{2} \)
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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

−11.14252836710623328309302902547, −10.71429403275157363091777006427, −9.522214844749970137144282123130, −8.627099282978081626866864417568, −7.82771492007906296111797569724, −6.60457291872571063147446886071, −5.90809165763560920929230583898, −5.21817018752778855393523661627, −3.53940349736192195917299334678, −2.02289646026153781692652573408, 0.49889328030731714675327017392, 1.82878947595064137707376966041, 3.70713513981330026908973685792, 4.69498145989414911705322550864, 5.73828643555855771972995918856, 6.94208864790942973755359468287, 8.018477126263870599768106484553, 8.863887599942858423913180385244, 9.624068024749601912218937366645, 10.78465942840951797706806978270

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