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Properties

Label 2-1470-35.13-c1-0-32
Degree $2$
Conductor $1470$
Sign $-0.566 + 0.824i$
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  + (0.707 − 0.707i)2-s + (−0.707 + 0.707i)3-s − 1.00i·4-s + (2.22 − 0.243i)5-s + 1.00i·6-s + (−0.707 − 0.707i)8-s − 1.00i·9-s + (1.39 − 1.74i)10-s − 5.33·11-s + (0.707 + 0.707i)12-s + (3.82 − 3.82i)13-s + (−1.39 + 1.74i)15-s − 1.00·16-s + (−5.19 − 5.19i)17-s + (−0.707 − 0.707i)18-s − 3.04·19-s + ⋯
L(s)  = 1  + (0.499 − 0.499i)2-s + (−0.408 + 0.408i)3-s − 0.500i·4-s + (0.994 − 0.108i)5-s + 0.408i·6-s + (−0.250 − 0.250i)8-s − 0.333i·9-s + (0.442 − 0.551i)10-s − 1.60·11-s + (0.204 + 0.204i)12-s + (1.06 − 1.06i)13-s + (−0.361 + 0.450i)15-s − 0.250·16-s + (−1.26 − 1.26i)17-s + (−0.166 − 0.166i)18-s − 0.697·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.566 + 0.824i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.566 + 0.824i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1470\)    =    \(2 \cdot 3 \cdot 5 \cdot 7^{2}\)
Sign: $-0.566 + 0.824i$
Analytic conductor: \(11.7380\)
Root analytic conductor: \(3.42607\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1470} (1273, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1470,\ (\ :1/2),\ -0.566 + 0.824i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.579057195\)
\(L(\frac12)\) \(\approx\) \(1.579057195\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 + 0.707i)T \)
3 \( 1 + (0.707 - 0.707i)T \)
5 \( 1 + (-2.22 + 0.243i)T \)
7 \( 1 \)
good11 \( 1 + 5.33T + 11T^{2} \)
13 \( 1 + (-3.82 + 3.82i)T - 13iT^{2} \)
17 \( 1 + (5.19 + 5.19i)T + 17iT^{2} \)
19 \( 1 + 3.04T + 19T^{2} \)
23 \( 1 + (1.63 + 1.63i)T + 23iT^{2} \)
29 \( 1 - 4.97iT - 29T^{2} \)
31 \( 1 + 6.37iT - 31T^{2} \)
37 \( 1 + (-2.11 + 2.11i)T - 37iT^{2} \)
41 \( 1 + 10.4iT - 41T^{2} \)
43 \( 1 + (-1.27 - 1.27i)T + 43iT^{2} \)
47 \( 1 + (-8.75 - 8.75i)T + 47iT^{2} \)
53 \( 1 + (4.91 + 4.91i)T + 53iT^{2} \)
59 \( 1 - 6.15T + 59T^{2} \)
61 \( 1 - 0.777iT - 61T^{2} \)
67 \( 1 + (5.72 - 5.72i)T - 67iT^{2} \)
71 \( 1 + 5.85T + 71T^{2} \)
73 \( 1 + (-5.03 + 5.03i)T - 73iT^{2} \)
79 \( 1 + 11.3iT - 79T^{2} \)
83 \( 1 + (-2.41 + 2.41i)T - 83iT^{2} \)
89 \( 1 - 3.19T + 89T^{2} \)
97 \( 1 + (4.72 + 4.72i)T + 97iT^{2} \)
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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

−9.344030072870565737077689345953, −8.710407030530354534553125624942, −7.57910152118073833017466473743, −6.42567713335701224869213380920, −5.69043032520078784842232236362, −5.13310730213984838891706183087, −4.24462424188308330777727986208, −2.93286453261007002752643181696, −2.21316676833868331495506079249, −0.52091996337644473069490293193, 1.73825733447043262473214833592, 2.60715483638275346859170034139, 4.05393178043500898610192175641, 4.93767161590473250960244530775, 5.91390397552145967197382808136, 6.31579420213726897768404260313, 7.08135285419764490166938248518, 8.229380418970045819144843569100, 8.691233948936327447990467252341, 9.836378503385035411372584865169

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