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Properties

Label 2-1470-7.2-c1-0-8
Degree $2$
Conductor $1470$
Sign $0.605 - 0.795i$
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  + (−0.5 + 0.866i)2-s + (−0.5 − 0.866i)3-s + (−0.499 − 0.866i)4-s + (0.5 − 0.866i)5-s + 0.999·6-s + 0.999·8-s + (−0.499 + 0.866i)9-s + (0.499 + 0.866i)10-s + (2.5 + 4.33i)11-s + (−0.499 + 0.866i)12-s + 5·13-s − 0.999·15-s + (−0.5 + 0.866i)16-s + (−2 − 3.46i)17-s + (−0.499 − 0.866i)18-s + (−3.5 + 6.06i)19-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.288 − 0.499i)3-s + (−0.249 − 0.433i)4-s + (0.223 − 0.387i)5-s + 0.408·6-s + 0.353·8-s + (−0.166 + 0.288i)9-s + (0.158 + 0.273i)10-s + (0.753 + 1.30i)11-s + (−0.144 + 0.249i)12-s + 1.38·13-s − 0.258·15-s + (−0.125 + 0.216i)16-s + (−0.485 − 0.840i)17-s + (−0.117 − 0.204i)18-s + (−0.802 + 1.39i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1470\)    =    \(2 \cdot 3 \cdot 5 \cdot 7^{2}\)
Sign: $0.605 - 0.795i$
Analytic conductor: \(11.7380\)
Root analytic conductor: \(3.42607\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1470} (961, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1470,\ (\ :1/2),\ 0.605 - 0.795i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.278284095\)
\(L(\frac12)\) \(\approx\) \(1.278284095\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 - 0.866i)T \)
3 \( 1 + (0.5 + 0.866i)T \)
5 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 \)
good11 \( 1 + (-2.5 - 4.33i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 5T + 13T^{2} \)
17 \( 1 + (2 + 3.46i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.5 - 6.06i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.5 - 0.866i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 + (1 + 1.73i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.5 - 0.866i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 5T + 41T^{2} \)
43 \( 1 - 12T + 43T^{2} \)
47 \( 1 + (5.5 - 9.52i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-4.5 - 7.79i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-2 - 3.46i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2 + 3.46i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6 - 10.3i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 2T + 71T^{2} \)
73 \( 1 + (-5 - 8.66i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-6 + 10.3i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 12T + 83T^{2} \)
89 \( 1 + (-7 + 12.1i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 8T + 97T^{2} \)
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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

−9.435413652965648313187719772280, −8.791737987112281731247924016495, −7.962443689353808899390997020851, −7.18129045237025270709502669029, −6.35665203711728041128358800612, −5.81845873209178267374812243227, −4.69336199533490211124736815670, −3.88286022290860931007901230824, −2.08686137011015217623153341833, −1.14721591490270083247109626258, 0.71917019474342350466976607913, 2.15224662097179625024985562554, 3.46912508939379291925484888133, 3.92594625838563655550085650694, 5.17787118292393023773268653176, 6.28784629431410555132294326749, 6.66259239372151572736551898856, 8.197063417870288484938099167247, 8.771450324102540866398319110555, 9.295540633667486553488677637269

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