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Properties

Label 2-490-7.4-c3-0-38
Degree $2$
Conductor $490$
Sign $-0.991 + 0.126i$
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  + (1 + 1.73i)2-s + (5 − 8.66i)3-s + (−1.99 + 3.46i)4-s + (−2.5 − 4.33i)5-s + 20·6-s − 7.99·8-s + (−36.5 − 63.2i)9-s + (5 − 8.66i)10-s + (−26.5 + 45.8i)11-s + (20 + 34.6i)12-s − 25·13-s − 50.0·15-s + (−8 − 13.8i)16-s + (7 − 12.1i)17-s + (73 − 126. i)18-s + (−47.5 − 82.2i)19-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (0.962 − 1.66i)3-s + (−0.249 + 0.433i)4-s + (−0.223 − 0.387i)5-s + 1.36·6-s − 0.353·8-s + (−1.35 − 2.34i)9-s + (0.158 − 0.273i)10-s + (−0.726 + 1.25i)11-s + (0.481 + 0.833i)12-s − 0.533·13-s − 0.860·15-s + (−0.125 − 0.216i)16-s + (0.0998 − 0.172i)17-s + (0.955 − 1.65i)18-s + (−0.573 − 0.993i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 + 0.126i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.991 + 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(490\)    =    \(2 \cdot 5 \cdot 7^{2}\)
Sign: $-0.991 + 0.126i$
Analytic conductor: \(28.9109\)
Root analytic conductor: \(5.37688\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{490} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 490,\ (\ :3/2),\ -0.991 + 0.126i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.163189309\)
\(L(\frac12)\) \(\approx\) \(1.163189309\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1 - 1.73i)T \)
5 \( 1 + (2.5 + 4.33i)T \)
7 \( 1 \)
good3 \( 1 + (-5 + 8.66i)T + (-13.5 - 23.3i)T^{2} \)
11 \( 1 + (26.5 - 45.8i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + 25T + 2.19e3T^{2} \)
17 \( 1 + (-7 + 12.1i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (47.5 + 82.2i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (0.5 + 0.866i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + 206T + 2.43e4T^{2} \)
31 \( 1 + (-54 + 93.5i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (-28.5 - 49.3i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + 243T + 6.89e4T^{2} \)
43 \( 1 - 434T + 7.95e4T^{2} \)
47 \( 1 + (115.5 + 200. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (131.5 - 227. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-12 + 20.7i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-58 - 100. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-102 + 176. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 484T + 3.57e5T^{2} \)
73 \( 1 + (346 - 599. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (233 + 403. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 228T + 5.71e5T^{2} \)
89 \( 1 + (181 + 313. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + 854T + 9.12e5T^{2} \)
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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

−9.711661750743829315605755925807, −8.886148322993568870120091960568, −7.999352762799497674328620029805, −7.35829880519985272699916093692, −6.81164520679210897722858169574, −5.58092264679079218858478342873, −4.32478136805557715532982590980, −2.85539040298633588255368351043, −1.92279758916243896096782582503, −0.25709458749117034338385768835, 2.32164983013291976961065628361, 3.27640923451254061856015313000, 3.91279655308931221229062052847, 5.01066332476458470956326969785, 5.88689582456059279781441253962, 7.77674634886274600532663943263, 8.505997016195774456057222541233, 9.394969239227515617716335388524, 10.23688378925989346792945135914, 10.77082902395692560891285584274

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