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Properties

Label 2-1470-21.20-c1-0-47
Degree $2$
Conductor $1470$
Sign $-0.158 + 0.987i$
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  − i·2-s + (1.67 − 0.449i)3-s − 4-s + 5-s + (−0.449 − 1.67i)6-s + i·8-s + (2.59 − 1.50i)9-s − i·10-s − 2.37i·11-s + (−1.67 + 0.449i)12-s − 3.07i·13-s + (1.67 − 0.449i)15-s + 16-s + 4.70·17-s + (−1.50 − 2.59i)18-s + 0.527i·19-s + ⋯
L(s)  = 1  − 0.707i·2-s + (0.965 − 0.259i)3-s − 0.5·4-s + 0.447·5-s + (−0.183 − 0.682i)6-s + 0.353i·8-s + (0.865 − 0.501i)9-s − 0.316i·10-s − 0.717i·11-s + (−0.482 + 0.129i)12-s − 0.852i·13-s + (0.431 − 0.116i)15-s + 0.250·16-s + 1.14·17-s + (−0.354 − 0.611i)18-s + 0.120i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.158 + 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.158 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1470\)    =    \(2 \cdot 3 \cdot 5 \cdot 7^{2}\)
Sign: $-0.158 + 0.987i$
Analytic conductor: \(11.7380\)
Root analytic conductor: \(3.42607\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1470} (881, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1470,\ (\ :1/2),\ -0.158 + 0.987i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.524555806\)
\(L(\frac12)\) \(\approx\) \(2.524555806\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
3 \( 1 + (-1.67 + 0.449i)T \)
5 \( 1 - T \)
7 \( 1 \)
good11 \( 1 + 2.37iT - 11T^{2} \)
13 \( 1 + 3.07iT - 13T^{2} \)
17 \( 1 - 4.70T + 17T^{2} \)
19 \( 1 - 0.527iT - 19T^{2} \)
23 \( 1 - 4.62iT - 23T^{2} \)
29 \( 1 + 0.405iT - 29T^{2} \)
31 \( 1 + 1.02iT - 31T^{2} \)
37 \( 1 + 5.54T + 37T^{2} \)
41 \( 1 + 8.61T + 41T^{2} \)
43 \( 1 - 11.5T + 43T^{2} \)
47 \( 1 + 8.82T + 47T^{2} \)
53 \( 1 + 12.2iT - 53T^{2} \)
59 \( 1 - 8.52T + 59T^{2} \)
61 \( 1 - 7.51iT - 61T^{2} \)
67 \( 1 - 13.4T + 67T^{2} \)
71 \( 1 + 11.7iT - 71T^{2} \)
73 \( 1 + 8.65iT - 73T^{2} \)
79 \( 1 - 1.91T + 79T^{2} \)
83 \( 1 + 12.9T + 83T^{2} \)
89 \( 1 + 6.18T + 89T^{2} \)
97 \( 1 - 19.6iT - 97T^{2} \)
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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

−9.434976373094418853309415041000, −8.466647463686024623599831748707, −7.987934602251247207239722972429, −7.02586733975365429483009364908, −5.86265320883396220311090356654, −5.08327093277394744445789946701, −3.64722222868610327919234733351, −3.21300693094106089932592092186, −2.08819268902074294730026066162, −0.992659046525693980701892426540, 1.56019316088906655391585713738, 2.72075385259154494641681900010, 3.87884922112012593270590877615, 4.68591680251974158737793508304, 5.56172824323235425171703057150, 6.75817218735708753844542890213, 7.23002299498050483489715939336, 8.247527048966207280882312399572, 8.782293230502514912049037055794, 9.751091453471996663375051572569

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