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Properties

Label 2-1470-35.9-c1-0-4
Degree $2$
Conductor $1470$
Sign $-0.742 - 0.669i$
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 + (1.23 + 1.86i)5-s + 0.999·6-s − 0.999i·8-s + (0.499 − 0.866i)9-s + (−0.133 − 2.23i)10-s + (1 + 1.73i)11-s + (−0.866 − 0.499i)12-s + 2i·13-s + (−2 − i)15-s + (−0.5 + 0.866i)16-s + (−6.92 + 4i)17-s + (−0.866 + 0.499i)18-s + (1 − 1.73i)19-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (−0.499 + 0.288i)3-s + (0.249 + 0.433i)4-s + (0.550 + 0.834i)5-s + 0.408·6-s − 0.353i·8-s + (0.166 − 0.288i)9-s + (−0.0423 − 0.705i)10-s + (0.301 + 0.522i)11-s + (−0.249 − 0.144i)12-s + 0.554i·13-s + (−0.516 − 0.258i)15-s + (−0.125 + 0.216i)16-s + (−1.68 + 0.970i)17-s + (−0.204 + 0.117i)18-s + (0.229 − 0.397i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.742 - 0.669i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.742 - 0.669i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1470\)    =    \(2 \cdot 3 \cdot 5 \cdot 7^{2}\)
Sign: $-0.742 - 0.669i$
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: \((2,\ 1470,\ (\ :1/2),\ -0.742 - 0.669i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6975768306\)
\(L(\frac12)\) \(\approx\) \(0.6975768306\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.23 - 1.86i)T \)
7 \( 1 \)
good11 \( 1 + (-1 - 1.73i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 2iT - 13T^{2} \)
17 \( 1 + (6.92 - 4i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1 + 1.73i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
31 \( 1 + (-3 - 5.19i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (6.92 + 4i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 6T + 41T^{2} \)
43 \( 1 + 8iT - 43T^{2} \)
47 \( 1 + (-3.46 - 2i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.73 + i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-4 - 6.92i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5 + 8.66i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (10.3 - 6i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 14T + 71T^{2} \)
73 \( 1 + (8.66 - 5i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2 + 3.46i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 16iT - 83T^{2} \)
89 \( 1 + (5 - 8.66i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 10iT - 97T^{2} \)
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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

−10.07293546696091619609531976085, −9.018519576796064579161470624757, −8.571223255449566076882297218437, −6.99198939715773763406630824027, −6.88347744942946440828938865205, −5.88905008694893698654373025265, −4.69489531455875546749186911112, −3.78871627499466656770142501303, −2.56572276467475078386482072280, −1.61853518653586458116299283376, 0.36478576322305608112807489077, 1.51117249862084215103890971719, 2.74776205274768922907896107161, 4.43750365018666826847535609725, 5.16946385134559538115146900612, 6.07818770476087779457636834260, 6.64786732032742371499355063447, 7.64483292351798004844644414050, 8.560410102716214747280720582487, 9.002226539030388836098638248613

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