Properties

Label 2-490-35.9-c1-0-8
Degree $2$
Conductor $490$
Sign $-0.330 - 0.943i$
Analytic cond. $3.91266$
Root an. cond. $1.97804$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.866 + 0.5i)2-s + (0.499 + 0.866i)4-s + (−0.133 + 2.23i)5-s + 0.999i·8-s + (−1.5 + 2.59i)9-s + (−1.23 + 1.86i)10-s + (−1.5 − 2.59i)11-s + 5i·13-s + (−0.5 + 0.866i)16-s + (−1.73 + i)17-s + (−2.59 + 1.5i)18-s + (2.5 − 4.33i)19-s + (−1.99 + 0.999i)20-s − 3i·22-s + (6.06 + 3.5i)23-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.249 + 0.433i)4-s + (−0.0599 + 0.998i)5-s + 0.353i·8-s + (−0.5 + 0.866i)9-s + (−0.389 + 0.590i)10-s + (−0.452 − 0.783i)11-s + 1.38i·13-s + (−0.125 + 0.216i)16-s + (−0.420 + 0.242i)17-s + (−0.612 + 0.353i)18-s + (0.573 − 0.993i)19-s + (−0.447 + 0.223i)20-s − 0.639i·22-s + (1.26 + 0.729i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.330 - 0.943i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.330 - 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(490\)    =    \(2 \cdot 5 \cdot 7^{2}\)
Sign: $-0.330 - 0.943i$
Analytic conductor: \(3.91266\)
Root analytic conductor: \(1.97804\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{490} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 490,\ (\ :1/2),\ -0.330 - 0.943i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.01718 + 1.43404i\)
\(L(\frac12)\) \(\approx\) \(1.01718 + 1.43404i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 - 0.5i)T \)
5 \( 1 + (0.133 - 2.23i)T \)
7 \( 1 \)
good3 \( 1 + (1.5 - 2.59i)T^{2} \)
11 \( 1 + (1.5 + 2.59i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 5iT - 13T^{2} \)
17 \( 1 + (1.73 - i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.5 + 4.33i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-6.06 - 3.5i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 4T + 29T^{2} \)
31 \( 1 + (1 + 1.73i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.866 - 0.5i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 3T + 41T^{2} \)
43 \( 1 - 2iT - 43T^{2} \)
47 \( 1 + (-6.06 - 3.5i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-7.79 + 4.5i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2 - 3.46i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3 + 5.19i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.73 - i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 6T + 71T^{2} \)
73 \( 1 + (-13.8 + 8i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-7 + 12.1i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 6iT - 83T^{2} \)
89 \( 1 + (1 - 1.73i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 12iT - 97T^{2} \)
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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

−11.20431354138504903109650998309, −10.74446310894925972923768427073, −9.369248047316683462904369112230, −8.390350618638631889355947715403, −7.35075183883875532309252744906, −6.67131875261339353571096160873, −5.62989690338705083596775032238, −4.62703886472197383041144228885, −3.30458746709726134917505246674, −2.35521198192347087251404604157, 0.896651004109062042349604060252, 2.66847739324534965920257532387, 3.84995930337431560083307506826, 5.03938790806980925456731354160, 5.64696656141744043016714341504, 6.91120559008235542510847857684, 8.074746369172733590685425481663, 8.956854497953787898374388494823, 9.912131235623781547540789915505, 10.71065327786033424938821959137

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