Properties

Label 2-490-245.103-c1-0-4
Degree $2$
Conductor $490$
Sign $0.491 - 0.870i$
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 + (−1.90 + 0.360i)3-s + (−0.294 − 0.955i)4-s + (−0.794 − 2.08i)5-s + (−0.841 + 1.74i)6-s + (−1.53 + 2.15i)7-s + (−0.943 − 0.330i)8-s + (0.704 − 0.276i)9-s + (−2.15 − 0.601i)10-s + (−1.64 + 4.18i)11-s + (0.905 + 1.71i)12-s + (4.26 + 0.480i)13-s + (0.823 + 2.51i)14-s + (2.26 + 3.69i)15-s + (−0.826 + 0.563i)16-s + (5.59 + 0.209i)17-s + ⋯
L(s)  = 1  + (0.419 − 0.568i)2-s + (−1.09 + 0.208i)3-s + (−0.147 − 0.477i)4-s + (−0.355 − 0.934i)5-s + (−0.343 + 0.712i)6-s + (−0.579 + 0.814i)7-s + (−0.333 − 0.116i)8-s + (0.234 − 0.0922i)9-s + (−0.681 − 0.190i)10-s + (−0.494 + 1.26i)11-s + (0.261 + 0.494i)12-s + (1.18 + 0.133i)13-s + (0.219 + 0.672i)14-s + (0.585 + 0.953i)15-s + (−0.206 + 0.140i)16-s + (1.35 + 0.0507i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.491 - 0.870i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.491 - 0.870i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(490\)    =    \(2 \cdot 5 \cdot 7^{2}\)
Sign: $0.491 - 0.870i$
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: \((2,\ 490,\ (\ :1/2),\ 0.491 - 0.870i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.572065 + 0.334047i\)
\(L(\frac12)\) \(\approx\) \(0.572065 + 0.334047i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.794 + 2.08i)T \)
7 \( 1 + (1.53 - 2.15i)T \)
good3 \( 1 + (1.90 - 0.360i)T + (2.79 - 1.09i)T^{2} \)
11 \( 1 + (1.64 - 4.18i)T + (-8.06 - 7.48i)T^{2} \)
13 \( 1 + (-4.26 - 0.480i)T + (12.6 + 2.89i)T^{2} \)
17 \( 1 + (-5.59 - 0.209i)T + (16.9 + 1.27i)T^{2} \)
19 \( 1 + (0.478 - 0.828i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.116 - 3.10i)T + (-22.9 + 1.71i)T^{2} \)
29 \( 1 + (5.51 - 1.25i)T + (26.1 - 12.5i)T^{2} \)
31 \( 1 + (7.07 - 4.08i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.29 - 1.21i)T + (20.8 - 30.5i)T^{2} \)
41 \( 1 + (-2.14 - 4.45i)T + (-25.5 + 32.0i)T^{2} \)
43 \( 1 + (-2.77 - 7.93i)T + (-33.6 + 26.8i)T^{2} \)
47 \( 1 + (0.364 + 0.269i)T + (13.8 + 44.9i)T^{2} \)
53 \( 1 + (-6.08 - 3.21i)T + (29.8 + 43.7i)T^{2} \)
59 \( 1 + (0.717 + 9.57i)T + (-58.3 + 8.79i)T^{2} \)
61 \( 1 + (1.77 - 5.75i)T + (-50.4 - 34.3i)T^{2} \)
67 \( 1 + (6.98 + 1.87i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (1.32 - 5.82i)T + (-63.9 - 30.8i)T^{2} \)
73 \( 1 + (13.3 - 9.87i)T + (21.5 - 69.7i)T^{2} \)
79 \( 1 + (8.01 + 4.62i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (2.03 + 18.0i)T + (-80.9 + 18.4i)T^{2} \)
89 \( 1 + (-4.26 - 10.8i)T + (-65.2 + 60.5i)T^{2} \)
97 \( 1 + (9.61 + 9.61i)T + 97iT^{2} \)
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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

−11.37936826523440925017763170329, −10.34971778088376485425316151626, −9.546182244465327549267222388672, −8.650395043386272943244590357868, −7.39837619057774230957579326958, −5.90743133974873570563611862438, −5.49733842400660400313436232772, −4.53697638598100966678974363059, −3.33755942800790092040644496329, −1.52589117923976534793132554108, 0.41661669661629420750399196835, 3.19262391784890253894866743376, 3.91100458441193847360378198219, 5.69209729647027229328048005207, 5.93877723330821876470831483497, 6.99929771304460654023441769677, 7.70425424815068156577040273177, 8.849625143149049341451080233657, 10.38022057943220434034656163568, 10.87586271907055831851173456824

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