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Properties

Label 2-1470-35.9-c1-0-20
Degree $2$
Conductor $1470$
Sign $-0.533 + 0.845i$
Analytic cond. $11.7380$
Root an. cond. $3.42607$
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 + (−0.866 + 0.5i)3-s + (0.499 + 0.866i)4-s + (−1.97 − 1.05i)5-s + 0.999·6-s − 0.999i·8-s + (0.499 − 0.866i)9-s + (1.18 + 1.89i)10-s + (1.52 + 2.64i)11-s + (−0.866 − 0.499i)12-s − 1.64i·13-s + (2.23 − 0.0743i)15-s + (−0.5 + 0.866i)16-s + (−2.51 + 1.45i)17-s + (−0.866 + 0.499i)18-s + (1.10 − 1.91i)19-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (−0.499 + 0.288i)3-s + (0.249 + 0.433i)4-s + (−0.882 − 0.470i)5-s + 0.408·6-s − 0.353i·8-s + (0.166 − 0.288i)9-s + (0.373 + 0.600i)10-s + (0.460 + 0.797i)11-s + (−0.249 − 0.144i)12-s − 0.455i·13-s + (0.577 − 0.0191i)15-s + (−0.125 + 0.216i)16-s + (−0.610 + 0.352i)17-s + (−0.204 + 0.117i)18-s + (0.253 − 0.439i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.533 + 0.845i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.533 + 0.845i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1470\)    =    \(2 \cdot 3 \cdot 5 \cdot 7^{2}\)
Sign: $-0.533 + 0.845i$
Analytic conductor: \(11.7380\)
Root analytic conductor: \(3.42607\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1470} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1470,\ (\ :1/2),\ -0.533 + 0.845i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4503125492\)
\(L(\frac12)\) \(\approx\) \(0.4503125492\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 + 0.5i)T \)
3 \( 1 + (0.866 - 0.5i)T \)
5 \( 1 + (1.97 + 1.05i)T \)
7 \( 1 \)
good11 \( 1 + (-1.52 - 2.64i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 1.64iT - 13T^{2} \)
17 \( 1 + (2.51 - 1.45i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.10 + 1.91i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.55 - 0.894i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 5.58T + 29T^{2} \)
31 \( 1 + (0.472 + 0.818i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-9.27 - 5.35i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 6.67T + 41T^{2} \)
43 \( 1 + 10.5iT - 43T^{2} \)
47 \( 1 + (9.85 + 5.68i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (5.01 - 2.89i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (2.98 + 5.17i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.222 - 0.386i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.09 + 0.632i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 14.8T + 71T^{2} \)
73 \( 1 + (6.85 - 3.95i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-7.14 + 12.3i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 0.874iT - 83T^{2} \)
89 \( 1 + (-8.75 + 15.1i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 10.4iT - 97T^{2} \)
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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

−9.276653216269095198081972871777, −8.582782550662089529124814715296, −7.63808846421599844808641191693, −7.05621912959305283528612031970, −5.98295728198998477448400834302, −4.82452068481102662026544258461, −4.16041726837390601547371339351, −3.14949063047664158267205750142, −1.65371199125510423703826343429, −0.28159895696496824558760632057, 1.10908180809278260501534475321, 2.63971201232348364452590442574, 3.85232009630184925728718859893, 4.82370898498095882620751932185, 6.06706003522145739657785271899, 6.51524906173907608591681893728, 7.51000782413578112269144921330, 7.936836638593752534102454385998, 8.983011992296085538540054144037, 9.601049364625148561846567945322

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