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Properties

Label 2-490-5.4-c3-0-27
Degree $2$
Conductor $490$
Sign $-0.0969 - 0.995i$
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 + 1.20i·3-s − 4·4-s + (11.1 − 1.08i)5-s − 2.40·6-s − 8i·8-s + 25.5·9-s + (2.16 + 22.2i)10-s − 26.4·11-s − 4.80i·12-s + 55.0i·13-s + (1.30 + 13.3i)15-s + 16·16-s − 49.4i·17-s + 51.1i·18-s − 5.46·19-s + ⋯
L(s)  = 1  + 0.707i·2-s + 0.230i·3-s − 0.5·4-s + (0.995 − 0.0969i)5-s − 0.163·6-s − 0.353i·8-s + 0.946·9-s + (0.0685 + 0.703i)10-s − 0.724·11-s − 0.115i·12-s + 1.17i·13-s + (0.0224 + 0.229i)15-s + 0.250·16-s − 0.705i·17-s + 0.669i·18-s − 0.0659·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0969 - 0.995i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.0969 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(490\)    =    \(2 \cdot 5 \cdot 7^{2}\)
Sign: $-0.0969 - 0.995i$
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: \((2,\ 490,\ (\ :3/2),\ -0.0969 - 0.995i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.287669991\)
\(L(\frac12)\) \(\approx\) \(2.287669991\)
\(L(\frac{5}{2})\) 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 \( 1 + (-11.1 + 1.08i)T \)
7 \( 1 \)
good3 \( 1 - 1.20iT - 27T^{2} \)
11 \( 1 + 26.4T + 1.33e3T^{2} \)
13 \( 1 - 55.0iT - 2.19e3T^{2} \)
17 \( 1 + 49.4iT - 4.91e3T^{2} \)
19 \( 1 + 5.46T + 6.85e3T^{2} \)
23 \( 1 - 1.07iT - 1.21e4T^{2} \)
29 \( 1 - 246.T + 2.43e4T^{2} \)
31 \( 1 - 228.T + 2.97e4T^{2} \)
37 \( 1 - 381. iT - 5.06e4T^{2} \)
41 \( 1 + 48.5T + 6.89e4T^{2} \)
43 \( 1 - 233. iT - 7.95e4T^{2} \)
47 \( 1 + 245. iT - 1.03e5T^{2} \)
53 \( 1 - 269. iT - 1.48e5T^{2} \)
59 \( 1 + 784.T + 2.05e5T^{2} \)
61 \( 1 - 425.T + 2.26e5T^{2} \)
67 \( 1 - 918. iT - 3.00e5T^{2} \)
71 \( 1 - 200.T + 3.57e5T^{2} \)
73 \( 1 + 778. iT - 3.89e5T^{2} \)
79 \( 1 + 189.T + 4.93e5T^{2} \)
83 \( 1 - 611. iT - 5.71e5T^{2} \)
89 \( 1 + 292.T + 7.04e5T^{2} \)
97 \( 1 - 1.42e3iT - 9.12e5T^{2} \)
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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

−10.36067649156932408752125359088, −9.882486716861842167260621514584, −9.049232854159549534108005817141, −8.078071350337794593025621448454, −6.89401660594816331015994752887, −6.33853457563550295545545803664, −5.02485432860333319675781908626, −4.46529083901990800050782841580, −2.74286210424236871857786727240, −1.25876270054082362311691489702, 0.823144472130852539112832917364, 2.05066307233339095894665573527, 3.07696013318141773430242355706, 4.51536174517572389770859437361, 5.52450504269409195506439723795, 6.50476332229592777622432099325, 7.72040802408378660870560574149, 8.601359329320629657527301240412, 9.791574104138100219393542637021, 10.30958034848972506865307480613

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