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Properties

Label 2-490-35.13-c1-0-12
Degree $2$
Conductor $490$
Sign $0.970 + 0.242i$
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  + (−0.707 + 0.707i)2-s + (2.05 − 2.05i)3-s − 1.00i·4-s + (0.830 + 2.07i)5-s + 2.90i·6-s + (0.707 + 0.707i)8-s − 5.44i·9-s + (−2.05 − 0.880i)10-s + 3.67·11-s + (−2.05 − 2.05i)12-s + (0.830 − 0.830i)13-s + (5.97 + 2.55i)15-s − 1.00·16-s + (−0.557 − 0.557i)17-s + (3.85 + 3.85i)18-s − 2.18·19-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s + (1.18 − 1.18i)3-s − 0.500i·4-s + (0.371 + 0.928i)5-s + 1.18i·6-s + (0.250 + 0.250i)8-s − 1.81i·9-s + (−0.649 − 0.278i)10-s + 1.10·11-s + (−0.593 − 0.593i)12-s + (0.230 − 0.230i)13-s + (1.54 + 0.660i)15-s − 0.250·16-s + (−0.135 − 0.135i)17-s + (0.908 + 0.908i)18-s − 0.502·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 + 0.242i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.970 + 0.242i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(490\)    =    \(2 \cdot 5 \cdot 7^{2}\)
Sign: $0.970 + 0.242i$
Analytic conductor: \(3.91266\)
Root analytic conductor: \(1.97804\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{490} (293, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 490,\ (\ :1/2),\ 0.970 + 0.242i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.77750 - 0.218553i\)
\(L(\frac12)\) \(\approx\) \(1.77750 - 0.218553i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 - 0.707i)T \)
5 \( 1 + (-0.830 - 2.07i)T \)
7 \( 1 \)
good3 \( 1 + (-2.05 + 2.05i)T - 3iT^{2} \)
11 \( 1 - 3.67T + 11T^{2} \)
13 \( 1 + (-0.830 + 0.830i)T - 13iT^{2} \)
17 \( 1 + (0.557 + 0.557i)T + 17iT^{2} \)
19 \( 1 + 2.18T + 19T^{2} \)
23 \( 1 + (-3.32 - 3.32i)T + 23iT^{2} \)
29 \( 1 + 2.62iT - 29T^{2} \)
31 \( 1 - 0.0415iT - 31T^{2} \)
37 \( 1 + (-0.181 + 0.181i)T - 37iT^{2} \)
41 \( 1 + 8.98iT - 41T^{2} \)
43 \( 1 + (0.474 + 0.474i)T + 43iT^{2} \)
47 \( 1 + (4.52 + 4.52i)T + 47iT^{2} \)
53 \( 1 + (-5.59 - 5.59i)T + 53iT^{2} \)
59 \( 1 + 10.7T + 59T^{2} \)
61 \( 1 + 1.99iT - 61T^{2} \)
67 \( 1 + (4.68 - 4.68i)T - 67iT^{2} \)
71 \( 1 - 8.11T + 71T^{2} \)
73 \( 1 + (6.97 - 6.97i)T - 73iT^{2} \)
79 \( 1 - 13.4iT - 79T^{2} \)
83 \( 1 + (9.73 - 9.73i)T - 83iT^{2} \)
89 \( 1 + 1.43T + 89T^{2} \)
97 \( 1 + (3.16 + 3.16i)T + 97iT^{2} \)
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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

−10.82905385974779501508846472866, −9.678811680292296132906892666946, −8.980700668151469588594291623223, −8.170310551967660360716354638157, −7.16955714603392127559101588046, −6.77582680669164951999607401777, −5.79484644507816567420568047687, −3.77648211474916582888124128782, −2.58728097240323325863422558624, −1.45246998753796732920901584994, 1.63450734209227672882924386958, 3.01494764491534141323671381010, 4.12586654939020593915613494079, 4.79276368231387018383880148425, 6.41159442938171269758197104164, 7.927639125678140530633184246925, 8.804783717335663936018712809602, 9.097628565559019606245679392925, 9.873791846216816286349243950357, 10.69391133796575059138448963510

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