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Properties

Label 2-1470-35.13-c1-0-9
Degree $2$
Conductor $1470$
Sign $0.438 - 0.898i$
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 + (1.62 + 1.53i)5-s + 1.00i·6-s + (−0.707 − 0.707i)8-s − 1.00i·9-s + (2.23 − 0.0614i)10-s − 4.55·11-s + (0.707 + 0.707i)12-s + (−1.77 + 1.77i)13-s + (−2.23 + 0.0614i)15-s − 1.00·16-s + (2.91 + 2.91i)17-s + (−0.707 − 0.707i)18-s + 3.77·19-s + ⋯
L(s)  = 1  + (0.499 − 0.499i)2-s + (−0.408 + 0.408i)3-s − 0.500i·4-s + (0.726 + 0.687i)5-s + 0.408i·6-s + (−0.250 − 0.250i)8-s − 0.333i·9-s + (0.706 − 0.0194i)10-s − 1.37·11-s + (0.204 + 0.204i)12-s + (−0.493 + 0.493i)13-s + (−0.577 + 0.0158i)15-s − 0.250·16-s + (0.707 + 0.707i)17-s + (−0.166 − 0.166i)18-s + 0.866·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.438 - 0.898i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.438 - 0.898i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1470\)    =    \(2 \cdot 3 \cdot 5 \cdot 7^{2}\)
Sign: $0.438 - 0.898i$
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: \((2,\ 1470,\ (\ :1/2),\ 0.438 - 0.898i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.733188722\)
\(L(\frac12)\) \(\approx\) \(1.733188722\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.62 - 1.53i)T \)
7 \( 1 \)
good11 \( 1 + 4.55T + 11T^{2} \)
13 \( 1 + (1.77 - 1.77i)T - 13iT^{2} \)
17 \( 1 + (-2.91 - 2.91i)T + 17iT^{2} \)
19 \( 1 - 3.77T + 19T^{2} \)
23 \( 1 + (-5.69 - 5.69i)T + 23iT^{2} \)
29 \( 1 + 1.55iT - 29T^{2} \)
31 \( 1 + 3.89iT - 31T^{2} \)
37 \( 1 + (8.08 - 8.08i)T - 37iT^{2} \)
41 \( 1 - 11.3iT - 41T^{2} \)
43 \( 1 + (-0.367 - 0.367i)T + 43iT^{2} \)
47 \( 1 + (-3.57 - 3.57i)T + 47iT^{2} \)
53 \( 1 + (5.96 + 5.96i)T + 53iT^{2} \)
59 \( 1 + 0.443T + 59T^{2} \)
61 \( 1 - 8.19iT - 61T^{2} \)
67 \( 1 + (-6.58 + 6.58i)T - 67iT^{2} \)
71 \( 1 + 6.68T + 71T^{2} \)
73 \( 1 + (3.07 - 3.07i)T - 73iT^{2} \)
79 \( 1 - 4.71iT - 79T^{2} \)
83 \( 1 + (-3.21 + 3.21i)T - 83iT^{2} \)
89 \( 1 - 6.04T + 89T^{2} \)
97 \( 1 + (0.462 + 0.462i)T + 97iT^{2} \)
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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

−9.927007206123534950444880893494, −9.264770871327018451468089906378, −7.933314866184406661838558578140, −7.09511970000706425058458019652, −6.12130715016971887218371638087, −5.37589311090189162279938722045, −4.80676818230533734291172597239, −3.42438861025527362593862578566, −2.78026955341600762302153520749, −1.50263906344486186987365440821, 0.61517790849048738271211631892, 2.24424214340535496351381495184, 3.20022255239804664785094883190, 4.83378643022662022221648446932, 5.24481001918419947763856803691, 5.79629208565959421744951551015, 7.02881394242097983975819630321, 7.49951026405161998109104694346, 8.474541348008337146081643193730, 9.210550185123692492768828263999

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