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Properties

Label 2-1470-35.4-c1-0-37
Degree $2$
Conductor $1470$
Sign $-0.407 + 0.913i$
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 + (−0.666 − 2.13i)5-s + 0.999·6-s − 0.999i·8-s + (0.499 + 0.866i)9-s + (−1.64 − 1.51i)10-s + (1.19 − 2.07i)11-s + (0.866 − 0.499i)12-s − 3.80i·13-s + (0.489 − 2.18i)15-s + (−0.5 − 0.866i)16-s + (−1.70 − 0.985i)17-s + (0.866 + 0.499i)18-s + (−2.08 − 3.61i)19-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.499 + 0.288i)3-s + (0.249 − 0.433i)4-s + (−0.298 − 0.954i)5-s + 0.408·6-s − 0.353i·8-s + (0.166 + 0.288i)9-s + (−0.520 − 0.479i)10-s + (0.360 − 0.624i)11-s + (0.249 − 0.144i)12-s − 1.05i·13-s + (0.126 − 0.563i)15-s + (−0.125 − 0.216i)16-s + (−0.413 − 0.238i)17-s + (0.204 + 0.117i)18-s + (−0.478 − 0.828i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.407 + 0.913i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.407 + 0.913i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1470\)    =    \(2 \cdot 3 \cdot 5 \cdot 7^{2}\)
Sign: $-0.407 + 0.913i$
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: \((2,\ 1470,\ (\ :1/2),\ -0.407 + 0.913i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.388595040\)
\(L(\frac12)\) \(\approx\) \(2.388595040\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.666 + 2.13i)T \)
7 \( 1 \)
good11 \( 1 + (-1.19 + 2.07i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 3.80iT - 13T^{2} \)
17 \( 1 + (1.70 + 0.985i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.08 + 3.61i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (7.07 - 4.08i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 9.16T + 29T^{2} \)
31 \( 1 + (0.803 - 1.39i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.09 + 0.631i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 3.19T + 41T^{2} \)
43 \( 1 + 4.90iT - 43T^{2} \)
47 \( 1 + (-2.60 + 1.50i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-10.5 - 6.08i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-6.52 + 11.2i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (6.54 + 11.3i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.00 - 2.88i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 6.94T + 71T^{2} \)
73 \( 1 + (0.438 + 0.253i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (6.21 + 10.7i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 1.89iT - 83T^{2} \)
89 \( 1 + (-0.989 - 1.71i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 12.0iT - 97T^{2} \)
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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

−9.224860673167958841497693496034, −8.485576478480528976552911882188, −7.87812422393805832308921576492, −6.70999203596875601193500259325, −5.68150448956130174617895955595, −4.93759071604686481328123344229, −4.07846059699434216406356321646, −3.30057075365151943580716841046, −2.15198683601766189502109718926, −0.70916863792238987319286859226, 1.92644833258041665061502438188, 2.73936800057701962502227201699, 4.04258548664873551339689430860, 4.31633031845063880277898221770, 5.93175907512918186938894624456, 6.61799986764424301735238037936, 7.11153023039977748352097243286, 8.091174398479741552827521329429, 8.665078700746013279008433280708, 9.881332485302271511905638570251

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