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Properties

Label 2-490-7.2-c3-0-3
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 + (−4 − 6.92i)3-s + (−1.99 − 3.46i)4-s + (−2.5 + 4.33i)5-s − 15.9·6-s − 7.99·8-s + (−18.4 + 32.0i)9-s + (5 + 8.66i)10-s + (−34 − 58.8i)11-s + (−15.9 + 27.7i)12-s − 34·13-s + 40·15-s + (−8 + 13.8i)16-s + (37 + 64.0i)17-s + (37 + 64.0i)18-s + (−64 + 110. i)19-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.769 − 1.33i)3-s + (−0.249 − 0.433i)4-s + (−0.223 + 0.387i)5-s − 1.08·6-s − 0.353·8-s + (−0.685 + 1.18i)9-s + (0.158 + 0.273i)10-s + (−0.931 − 1.61i)11-s + (−0.384 + 0.666i)12-s − 0.725·13-s + 0.688·15-s + (−0.125 + 0.216i)16-s + (0.527 + 0.914i)17-s + (0.484 + 0.839i)18-s + (−0.772 + 1.33i)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: $\chi_{490} (471, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 490,\ (\ :3/2),\ 0.968 - 0.250i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.3676996589\)
\(L(\frac12)\) \(\approx\) \(0.3676996589\)
\(L(\frac{5}{2})\) 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 \( 1 + (4 + 6.92i)T + (-13.5 + 23.3i)T^{2} \)
11 \( 1 + (34 + 58.8i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + 34T + 2.19e3T^{2} \)
17 \( 1 + (-37 - 64.0i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (64 - 110. i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-40 + 69.2i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 - 286T + 2.43e4T^{2} \)
31 \( 1 + (12 + 20.7i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (147 - 254. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + 66T + 6.89e4T^{2} \)
43 \( 1 + 124T + 7.95e4T^{2} \)
47 \( 1 + (-156 + 270. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (-17 - 29.4i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (-84 - 145. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-85 + 147. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (282 + 488. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 616T + 3.57e5T^{2} \)
73 \( 1 + (-125 - 216. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (-472 + 817. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + 672T + 5.71e5T^{2} \)
89 \( 1 + (715 - 1.23e3i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 - 1.27e3T + 9.12e5T^{2} \)
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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

−10.61700530524657880433539397650, −10.32139278051352240591238492224, −8.438627845440572139599807026080, −7.943052278604753108627977826230, −6.62431362090390501237188221767, −6.02179859419833520947823418522, −5.07995387512281360000993068421, −3.48485678526724594609695455488, −2.35302027481907729330631931392, −1.01558961997591997477331378189, 0.13372096065715616314592159822, 2.71209040174578898634372992446, 4.25435309721528595591569604460, 4.90603317107726189151037830979, 5.31955511391296798737809022707, 6.81587846366926063633504034523, 7.58646586992676928057198763710, 8.889942273013904019319057060852, 9.733084948197398240755343733818, 10.35029174515526122436598760404

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