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Properties

Label 2-1470-105.104-c1-0-10
Degree $2$
Conductor $1470$
Sign $0.679 - 0.733i$
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  + 2-s + (−1.17 − 1.27i)3-s + 4-s + (−1.07 − 1.95i)5-s + (−1.17 − 1.27i)6-s + 8-s + (−0.259 + 2.98i)9-s + (−1.07 − 1.95i)10-s + 6.01i·11-s + (−1.17 − 1.27i)12-s − 0.864·13-s + (−1.23 + 3.67i)15-s + 16-s + 2.84i·17-s + (−0.259 + 2.98i)18-s + 7.12i·19-s + ⋯
L(s)  = 1  + 0.707·2-s + (−0.675 − 0.737i)3-s + 0.5·4-s + (−0.482 − 0.875i)5-s + (−0.477 − 0.521i)6-s + 0.353·8-s + (−0.0865 + 0.996i)9-s + (−0.341 − 0.619i)10-s + 1.81i·11-s + (−0.337 − 0.368i)12-s − 0.239·13-s + (−0.319 + 0.947i)15-s + 0.250·16-s + 0.690i·17-s + (−0.0612 + 0.704i)18-s + 1.63i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.679 - 0.733i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.679 - 0.733i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1470\)    =    \(2 \cdot 3 \cdot 5 \cdot 7^{2}\)
Sign: $0.679 - 0.733i$
Analytic conductor: \(11.7380\)
Root analytic conductor: \(3.42607\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1470} (1469, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1470,\ (\ :1/2),\ 0.679 - 0.733i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.394045207\)
\(L(\frac12)\) \(\approx\) \(1.394045207\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 + (1.17 + 1.27i)T \)
5 \( 1 + (1.07 + 1.95i)T \)
7 \( 1 \)
good11 \( 1 - 6.01iT - 11T^{2} \)
13 \( 1 + 0.864T + 13T^{2} \)
17 \( 1 - 2.84iT - 17T^{2} \)
19 \( 1 - 7.12iT - 19T^{2} \)
23 \( 1 + 2.10T + 23T^{2} \)
29 \( 1 + 9.40iT - 29T^{2} \)
31 \( 1 - 2.68iT - 31T^{2} \)
37 \( 1 - 6.81iT - 37T^{2} \)
41 \( 1 - 5.32T + 41T^{2} \)
43 \( 1 + 2.68iT - 43T^{2} \)
47 \( 1 + 3.12iT - 47T^{2} \)
53 \( 1 - 11.9T + 53T^{2} \)
59 \( 1 + 7.09T + 59T^{2} \)
61 \( 1 - 9.33iT - 61T^{2} \)
67 \( 1 - 6.56iT - 67T^{2} \)
71 \( 1 - 4.46iT - 71T^{2} \)
73 \( 1 + 12.7T + 73T^{2} \)
79 \( 1 - 3.35T + 79T^{2} \)
83 \( 1 + 1.28iT - 83T^{2} \)
89 \( 1 + 3.08T + 89T^{2} \)
97 \( 1 + 1.90T + 97T^{2} \)
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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

−9.929388566736466799888845865720, −8.529961022867498077524855551068, −7.73693619588755488137752019481, −7.24391236461117859022806336844, −6.17274810601233107322088642870, −5.49778368513014793623874353035, −4.53874836422940838471742540852, −4.01389764003041488281674538854, −2.25956707157294622327476123466, −1.40101984393496164137378553040, 0.49060934249954040486165939211, 2.77448113482149614998792877239, 3.37550221636712908888379153633, 4.30923021377724184616832979909, 5.24333140675222981536181818198, 6.00729280757795254495647431412, 6.74915806790815420029116793758, 7.53023183700026379880809758355, 8.723188656190005233024928364505, 9.453914867123692808502470009634

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