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Properties

Label 2-1470-35.4-c1-0-0
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 − 6i·13-s + (−2 + i)15-s + (−0.5 − 0.866i)16-s + (−3.46 − 2i)17-s + (−0.866 − 0.499i)18-s + (3 + 5.19i)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 − 1.66i·13-s + (−0.516 + 0.258i)15-s + (−0.125 − 0.216i)16-s + (−0.840 − 0.485i)17-s + (−0.204 − 0.117i)18-s + (0.688 + 1.19i)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: $\chi_{1470} (949, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1470,\ (\ :1/2),\ -0.742 + 0.669i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1803055655\)
\(L(\frac12)\) \(\approx\) \(0.1803055655\)
\(L(\frac{3}{2})\) 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 \( 1 + 6iT - 13T^{2} \)
17 \( 1 + (3.46 + 2i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3 - 5.19i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (6.92 - 4i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 6T + 29T^{2} \)
31 \( 1 + (1 - 1.73i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (3.46 - 2i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 2T + 41T^{2} \)
43 \( 1 - 4iT - 43T^{2} \)
47 \( 1 + (-6.92 + 4i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (5.19 + 3i)T + (26.5 + 45.8i)T^{2} \)
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 \( 1 + (-6.92 - 4i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 6T + 71T^{2} \)
73 \( 1 + (12.1 + 7i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (6 + 10.3i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 8iT - 83T^{2} \)
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

−10.08403421110949839698191972845, −9.244410583857898013952538950184, −8.068670858839786159990322018057, −7.82439017538853286532198362637, −7.07976076336612739762800630640, −5.98663250134332642708621498983, −5.16306120676291996009026296036, −3.85002376590902417946788670153, −3.07249282525364232917865928460, −1.92193422188403187054799705015, 0.07870159420980363761312125416, 1.56408694582315674107207489478, 2.52826816062252476925996162429, 3.89021146762251261892014424131, 4.43864949452672669654151337814, 5.79420328283620025578242233687, 6.89619725752307381100975145102, 7.53841579886929693253077480187, 8.476751925152905246427794698442, 8.946436508395661309287165888913

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