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Properties

Label 2-490-5.4-c3-0-43
Degree $2$
Conductor $490$
Sign $0.640 + 0.767i$
Analytic cond. $28.9109$
Root an. cond. $5.37688$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2i·2-s + 9.18i·3-s − 4·4-s + (8.58 − 7.16i)5-s + 18.3·6-s + 8i·8-s − 57.4·9-s + (−14.3 − 17.1i)10-s + 35.4·11-s − 36.7i·12-s − 45.5i·13-s + (65.7 + 78.8i)15-s + 16·16-s − 93.7i·17-s + 114. i·18-s − 44.8·19-s + ⋯
L(s)  = 1  − 0.707i·2-s + 1.76i·3-s − 0.5·4-s + (0.767 − 0.640i)5-s + 1.25·6-s + 0.353i·8-s − 2.12·9-s + (−0.452 − 0.542i)10-s + 0.972·11-s − 0.884i·12-s − 0.971i·13-s + (1.13 + 1.35i)15-s + 0.250·16-s − 1.33i·17-s + 1.50i·18-s − 0.541·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.640 + 0.767i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.640 + 0.767i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(490\)    =    \(2 \cdot 5 \cdot 7^{2}\)
Sign: $0.640 + 0.767i$
Analytic conductor: \(28.9109\)
Root analytic conductor: \(5.37688\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{490} (99, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 490,\ (\ :3/2),\ 0.640 + 0.767i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.851618660\)
\(L(\frac12)\) \(\approx\) \(1.851618660\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 2iT \)
5 \( 1 + (-8.58 + 7.16i)T \)
7 \( 1 \)
good3 \( 1 - 9.18iT - 27T^{2} \)
11 \( 1 - 35.4T + 1.33e3T^{2} \)
13 \( 1 + 45.5iT - 2.19e3T^{2} \)
17 \( 1 + 93.7iT - 4.91e3T^{2} \)
19 \( 1 + 44.8T + 6.85e3T^{2} \)
23 \( 1 + 122. iT - 1.21e4T^{2} \)
29 \( 1 - 17.8T + 2.43e4T^{2} \)
31 \( 1 - 27.1T + 2.97e4T^{2} \)
37 \( 1 - 78.0iT - 5.06e4T^{2} \)
41 \( 1 + 21.0T + 6.89e4T^{2} \)
43 \( 1 - 467. iT - 7.95e4T^{2} \)
47 \( 1 + 578. iT - 1.03e5T^{2} \)
53 \( 1 + 161. iT - 1.48e5T^{2} \)
59 \( 1 - 559.T + 2.05e5T^{2} \)
61 \( 1 + 108.T + 2.26e5T^{2} \)
67 \( 1 + 407. iT - 3.00e5T^{2} \)
71 \( 1 - 1.15e3T + 3.57e5T^{2} \)
73 \( 1 - 256. iT - 3.89e5T^{2} \)
79 \( 1 - 853.T + 4.93e5T^{2} \)
83 \( 1 + 828. iT - 5.71e5T^{2} \)
89 \( 1 + 164.T + 7.04e5T^{2} \)
97 \( 1 + 38.6iT - 9.12e5T^{2} \)
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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

−10.20829348904250037368845109121, −9.746189698323849268673788920559, −8.996178297742509161045545278033, −8.332321131577115220660343211813, −6.36326088712521724125768410951, −5.20341888658631454898270478713, −4.66762922020813341484945247198, −3.60939256838860643562906166619, −2.49636112891657591382587486649, −0.62330824837801061923975043690, 1.30803328998642762290393678325, 2.17592369423845481583254901411, 3.79603798907813911829537327936, 5.59980237675550095003645935641, 6.39993910450492563613749261469, 6.79060303349807134399765825821, 7.68889286727832647363416114731, 8.672429057432562558208416783345, 9.467489550749960344773942933677, 10.79937414799034011323665936848

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