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Properties

Label 2-1470-7.2-c1-0-9
Degree $2$
Conductor $1470$
Sign $0.827 + 0.561i$
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 + (3.12 + 5.40i)11-s + (−0.499 + 0.866i)12-s + 5.65·13-s − 0.999·15-s + (−0.5 + 0.866i)16-s + (2.70 + 4.68i)17-s + (0.499 + 0.866i)18-s + (−0.585 + 1.01i)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 + (0.941 + 1.63i)11-s + (−0.144 + 0.249i)12-s + 1.56·13-s − 0.258·15-s + (−0.125 + 0.216i)16-s + (0.656 + 1.13i)17-s + (0.117 + 0.204i)18-s + (−0.134 + 0.232i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.827 + 0.561i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.827 + 0.561i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1470\)    =    \(2 \cdot 3 \cdot 5 \cdot 7^{2}\)
Sign: $0.827 + 0.561i$
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: \((2,\ 1470,\ (\ :1/2),\ 0.827 + 0.561i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.021313619\)
\(L(\frac12)\) \(\approx\) \(2.021313619\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.12 - 5.40i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 5.65T + 13T^{2} \)
17 \( 1 + (-2.70 - 4.68i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.585 - 1.01i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (4.41 - 7.64i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 5.41T + 29T^{2} \)
31 \( 1 + (0.878 + 1.52i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (4.12 - 7.13i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 6.48T + 41T^{2} \)
43 \( 1 + 5.07T + 43T^{2} \)
47 \( 1 + (-3.12 + 5.40i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (5.82 + 10.0i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (4.41 + 7.64i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.171 - 0.297i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.53 - 4.39i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 12.4T + 71T^{2} \)
73 \( 1 + (1 + 1.73i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (4 - 6.92i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 10.8T + 83T^{2} \)
89 \( 1 + (-2.41 + 4.18i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 10.4T + 97T^{2} \)
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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

−9.713920634856057087413992861116, −8.627743576665876264463318049398, −7.941453384254639528342733455482, −6.76165846254131705935828701383, −6.12081381959912215902003974962, −5.27987867562086044585694651649, −4.19552560235778868955162520118, −3.52095107928326834979176602709, −1.80945408913290977708976628204, −1.39545129326193535725249508007, 0.863157317403497290581477272596, 2.89262780006446531499050614868, 3.67695180169675040562520230625, 4.52068729723793380849326453846, 5.77213848662281517150752981442, 6.11618100169731134753731770725, 6.85993177000168409139113180882, 8.082622224248595587465726056622, 8.766134986896067524332111179747, 9.349051444121038015040972061775

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