Properties

Label 2-490-7.4-c3-0-28
Degree $2$
Conductor $490$
Sign $0.968 + 0.250i$
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 + (−0.5 + 0.866i)3-s + (−1.99 + 3.46i)4-s + (−2.5 − 4.33i)5-s − 1.99·6-s − 7.99·8-s + (13 + 22.5i)9-s + (5 − 8.66i)10-s + (32.5 − 56.2i)11-s + (−1.99 − 3.46i)12-s − 13·13-s + 5·15-s + (−8 − 13.8i)16-s + (−36.5 + 63.2i)17-s + (−26 + 45.0i)18-s + (−71 − 122. i)19-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.0962 + 0.166i)3-s + (−0.249 + 0.433i)4-s + (−0.223 − 0.387i)5-s − 0.136·6-s − 0.353·8-s + (0.481 + 0.833i)9-s + (0.158 − 0.273i)10-s + (0.890 − 1.54i)11-s + (−0.0481 − 0.0833i)12-s − 0.277·13-s + 0.0860·15-s + (−0.125 − 0.216i)16-s + (−0.520 + 0.901i)17-s + (−0.340 + 0.589i)18-s + (−0.857 − 1.48i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 + 0.250i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.968 + 0.250i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(490\)    =    \(2 \cdot 5 \cdot 7^{2}\)
Sign: $0.968 + 0.250i$
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: \((2,\ 490,\ (\ :3/2),\ 0.968 + 0.250i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.889576136\)
\(L(\frac12)\) \(\approx\) \(1.889576136\)
\(L(\frac{5}{2})\) 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 \( 1 + (0.5 - 0.866i)T + (-13.5 - 23.3i)T^{2} \)
11 \( 1 + (-32.5 + 56.2i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + 13T + 2.19e3T^{2} \)
17 \( 1 + (36.5 - 63.2i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (71 + 122. i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (65 + 112. i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 - 111T + 2.43e4T^{2} \)
31 \( 1 + (-128 + 221. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (-133 - 230. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 - 424T + 6.89e4T^{2} \)
43 \( 1 - 534T + 7.95e4T^{2} \)
47 \( 1 + (134.5 + 232. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-66 + 114. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (112 - 193. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (286 + 495. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-54 + 93.5i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 560T + 3.57e5T^{2} \)
73 \( 1 + (-293 + 507. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (28.5 + 49.3i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 252T + 5.71e5T^{2} \)
89 \( 1 + (92 + 159. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 - 605T + 9.12e5T^{2} \)
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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

−10.71905748337648356764883439327, −9.396651513163589254807766324706, −8.524956359291930873564853566777, −7.921001573905990565691301891935, −6.60150933786125653869594006275, −5.97641792145944832709074307691, −4.62645578726121500925226115625, −4.09743646143407510595147999695, −2.50998237906812847566599580845, −0.59867850325399758163217112548, 1.22319812299503930975887108131, 2.42463756231223284749520522659, 3.89474803270797931561734817209, 4.46685828850079920784737400757, 5.97868584140683669841297859416, 6.85221205693356790276005297288, 7.68198438375602298811464158386, 9.199576984892554158845589875269, 9.728569909170889564123770851488, 10.60752649735668404294042642443

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