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Properties

Label 2-1470-105.104-c1-0-41
Degree $2$
Conductor $1470$
Sign $0.402 + 0.915i$
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  + 2-s + (−1.68 − 0.396i)3-s + 4-s + (−2.18 − 0.469i)5-s + (−1.68 − 0.396i)6-s + 8-s + (2.68 + 1.33i)9-s + (−2.18 − 0.469i)10-s − 0.939i·11-s + (−1.68 − 0.396i)12-s + 2·13-s + (3.5 + 1.65i)15-s + 16-s + 6.63i·17-s + (2.68 + 1.33i)18-s − 3.46i·19-s + ⋯
L(s)  = 1  + 0.707·2-s + (−0.973 − 0.228i)3-s + 0.5·4-s + (−0.977 − 0.210i)5-s + (−0.688 − 0.161i)6-s + 0.353·8-s + (0.895 + 0.445i)9-s + (−0.691 − 0.148i)10-s − 0.283i·11-s + (−0.486 − 0.114i)12-s + 0.554·13-s + (0.903 + 0.428i)15-s + 0.250·16-s + 1.60i·17-s + (0.633 + 0.314i)18-s − 0.794i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.402 + 0.915i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.402 + 0.915i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1470\)    =    \(2 \cdot 3 \cdot 5 \cdot 7^{2}\)
Sign: $0.402 + 0.915i$
Analytic conductor: \(11.7380\)
Root analytic conductor: \(3.42607\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1470} (1469, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1470,\ (\ :1/2),\ 0.402 + 0.915i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.440458437\)
\(L(\frac12)\) \(\approx\) \(1.440458437\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 + (1.68 + 0.396i)T \)
5 \( 1 + (2.18 + 0.469i)T \)
7 \( 1 \)
good11 \( 1 + 0.939iT - 11T^{2} \)
13 \( 1 - 2T + 13T^{2} \)
17 \( 1 - 6.63iT - 17T^{2} \)
19 \( 1 + 3.46iT - 19T^{2} \)
23 \( 1 + 1.37T + 23T^{2} \)
29 \( 1 + 3.31iT - 29T^{2} \)
31 \( 1 + 7.57iT - 31T^{2} \)
37 \( 1 + 8.21iT - 37T^{2} \)
41 \( 1 - 7.37T + 41T^{2} \)
43 \( 1 - 1.08iT - 43T^{2} \)
47 \( 1 + 8.51iT - 47T^{2} \)
53 \( 1 + 4.37T + 53T^{2} \)
59 \( 1 - 13.1T + 59T^{2} \)
61 \( 1 + 12.7iT - 61T^{2} \)
67 \( 1 + 2.37iT - 67T^{2} \)
71 \( 1 + 8.51iT - 71T^{2} \)
73 \( 1 - 2T + 73T^{2} \)
79 \( 1 - 9.11T + 79T^{2} \)
83 \( 1 - 11.8iT - 83T^{2} \)
89 \( 1 - 1.37T + 89T^{2} \)
97 \( 1 - 15.1T + 97T^{2} \)
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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

−9.417621584867154247946132335813, −8.242043210671983289130489574299, −7.69190882834725340886257179947, −6.69336109825476359187973072325, −6.03653712955260084396071964921, −5.22768538550105004708537002910, −4.18506998273274467193325382939, −3.73220431153093608519635794140, −2.10097279074241338731826608589, −0.61933844343223262848635918801, 1.12875836810330437357828343328, 2.91234029814802367624063386089, 3.85708842052351942088757812615, 4.65371287672185367552504898588, 5.35260995434631783874183166842, 6.37247462943644690668193265909, 7.05634124451526966139205830316, 7.74158427182372129692836029585, 8.852668583432075552247538891722, 9.930501813620624360827747709750

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