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Properties

Label 2-490-5.4-c3-0-8
Degree $2$
Conductor $490$
Sign $0.954 - 0.298i$
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 − 4.49i·3-s − 4·4-s + (−3.34 − 10.6i)5-s − 8.98·6-s + 8i·8-s + 6.79·9-s + (−21.3 + 6.68i)10-s + 13.9·11-s + 17.9i·12-s + 78.0i·13-s + (−47.9 + 15.0i)15-s + 16·16-s + 105. i·17-s − 13.5i·18-s − 152.·19-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.865i·3-s − 0.5·4-s + (−0.298 − 0.954i)5-s − 0.611·6-s + 0.353i·8-s + 0.251·9-s + (−0.674 + 0.211i)10-s + 0.381·11-s + 0.432i·12-s + 1.66i·13-s + (−0.825 + 0.258i)15-s + 0.250·16-s + 1.49i·17-s − 0.177i·18-s − 1.83·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.954 - 0.298i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.954 - 0.298i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(490\)    =    \(2 \cdot 5 \cdot 7^{2}\)
Sign: $0.954 - 0.298i$
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: \((2,\ 490,\ (\ :3/2),\ 0.954 - 0.298i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.8388878371\)
\(L(\frac12)\) \(\approx\) \(0.8388878371\)
\(L(\frac{5}{2})\) 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 \( 1 + (3.34 + 10.6i)T \)
7 \( 1 \)
good3 \( 1 + 4.49iT - 27T^{2} \)
11 \( 1 - 13.9T + 1.33e3T^{2} \)
13 \( 1 - 78.0iT - 2.19e3T^{2} \)
17 \( 1 - 105. iT - 4.91e3T^{2} \)
19 \( 1 + 152.T + 6.85e3T^{2} \)
23 \( 1 - 108. iT - 1.21e4T^{2} \)
29 \( 1 - 109.T + 2.43e4T^{2} \)
31 \( 1 + 197.T + 2.97e4T^{2} \)
37 \( 1 - 254. iT - 5.06e4T^{2} \)
41 \( 1 - 193.T + 6.89e4T^{2} \)
43 \( 1 - 41.5iT - 7.95e4T^{2} \)
47 \( 1 + 109. iT - 1.03e5T^{2} \)
53 \( 1 + 11.0iT - 1.48e5T^{2} \)
59 \( 1 + 192.T + 2.05e5T^{2} \)
61 \( 1 + 34.9T + 2.26e5T^{2} \)
67 \( 1 - 374. iT - 3.00e5T^{2} \)
71 \( 1 + 575.T + 3.57e5T^{2} \)
73 \( 1 + 522. iT - 3.89e5T^{2} \)
79 \( 1 - 456.T + 4.93e5T^{2} \)
83 \( 1 - 773. iT - 5.71e5T^{2} \)
89 \( 1 + 1.32e3T + 7.04e5T^{2} \)
97 \( 1 + 1.03e3iT - 9.12e5T^{2} \)
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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

−10.75807750848323653799023313691, −9.623346967228419210206432241137, −8.780321596656030686087541659826, −8.110109564044803470368852751632, −6.91994621186228711483027868788, −6.02186240355446664435530711891, −4.47775193392808094529089537621, −3.94147586785282785797352148381, −1.94007148846914004131253206323, −1.38786331496827651559013011527, 0.27051402875385728498573809843, 2.70453877534792129798822629077, 3.84347845378550479482251247918, 4.73678156688487067680218569924, 5.87414813134759199310932896807, 6.87861473471059313160846391245, 7.66031863296838379415371225409, 8.699606820923020784948749262814, 9.652270846591749829778790573874, 10.55650338056346032493956234801

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