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Properties

Label 2-1470-105.104-c1-0-2
Degree $2$
Conductor $1470$
Sign $-0.938 - 0.345i$
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 + (−1.18 + 1.26i)3-s + 4-s + (0.686 − 2.12i)5-s + (1.18 − 1.26i)6-s − 8-s + (−0.186 − 2.99i)9-s + (−0.686 + 2.12i)10-s + 4.25i·11-s + (−1.18 + 1.26i)12-s − 2·13-s + (1.87 + 3.39i)15-s + 16-s − 6.63i·17-s + (0.186 + 2.99i)18-s + 3.46i·19-s + ⋯
L(s)  = 1  − 0.707·2-s + (−0.684 + 0.728i)3-s + 0.5·4-s + (0.306 − 0.951i)5-s + (0.484 − 0.515i)6-s − 0.353·8-s + (−0.0620 − 0.998i)9-s + (−0.216 + 0.672i)10-s + 1.28i·11-s + (−0.342 + 0.364i)12-s − 0.554·13-s + (0.483 + 0.875i)15-s + 0.250·16-s − 1.60i·17-s + (0.0438 + 0.705i)18-s + 0.794i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.938 - 0.345i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.938 - 0.345i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1470\)    =    \(2 \cdot 3 \cdot 5 \cdot 7^{2}\)
Sign: $-0.938 - 0.345i$
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: \((2,\ 1470,\ (\ :1/2),\ -0.938 - 0.345i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2220733284\)
\(L(\frac12)\) \(\approx\) \(0.2220733284\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.18 - 1.26i)T \)
5 \( 1 + (-0.686 + 2.12i)T \)
7 \( 1 \)
good11 \( 1 - 4.25iT - 11T^{2} \)
13 \( 1 + 2T + 13T^{2} \)
17 \( 1 + 6.63iT - 17T^{2} \)
19 \( 1 - 3.46iT - 19T^{2} \)
23 \( 1 + 4.37T + 23T^{2} \)
29 \( 1 + 3.31iT - 29T^{2} \)
31 \( 1 + 2.37iT - 31T^{2} \)
37 \( 1 - 11.6iT - 37T^{2} \)
41 \( 1 - 1.62T + 41T^{2} \)
43 \( 1 - 11.0iT - 43T^{2} \)
47 \( 1 + 1.87iT - 47T^{2} \)
53 \( 1 + 1.37T + 53T^{2} \)
59 \( 1 + 4.11T + 59T^{2} \)
61 \( 1 - 2.81iT - 61T^{2} \)
67 \( 1 - 7.57iT - 67T^{2} \)
71 \( 1 - 1.87iT - 71T^{2} \)
73 \( 1 + 2T + 73T^{2} \)
79 \( 1 + 8.11T + 79T^{2} \)
83 \( 1 + 1.43iT - 83T^{2} \)
89 \( 1 + 4.37T + 89T^{2} \)
97 \( 1 - 2.11T + 97T^{2} \)
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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

−9.648304787318235715115791995647, −9.550235309477905716059747125079, −8.367950991483840228574644707924, −7.55618680054658418354337312647, −6.60435305875096911166052674978, −5.69531783763813326335933050794, −4.83383657306777843223296569265, −4.24172254339516635578841387428, −2.67371401790238602243684260157, −1.34030470561268599264958109619, 0.12481520335432469099098900151, 1.68198667446727389748374353355, 2.63577472445902719088369971702, 3.81415785378694642388076338441, 5.45337106132197789571519096472, 6.04220996845439553859974218777, 6.74476466543811615451028628138, 7.48810253907269644213764242285, 8.249487442728807659437592636271, 9.074822956158802701908069790245

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