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Properties

Label 2-1470-21.20-c1-0-5
Degree $2$
Conductor $1470$
Sign $-0.568 - 0.822i$
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 + (0.431 − 1.67i)3-s − 4-s − 5-s + (1.67 + 0.431i)6-s − i·8-s + (−2.62 − 1.44i)9-s − i·10-s + 5.13i·11-s + (−0.431 + 1.67i)12-s − 5.00i·13-s + (−0.431 + 1.67i)15-s + 16-s − 3.75·17-s + (1.44 − 2.62i)18-s + 2.69i·19-s + ⋯
L(s)  = 1  + 0.707i·2-s + (0.249 − 0.968i)3-s − 0.5·4-s − 0.447·5-s + (0.684 + 0.176i)6-s − 0.353i·8-s + (−0.875 − 0.482i)9-s − 0.316i·10-s + 1.54i·11-s + (−0.124 + 0.484i)12-s − 1.38i·13-s + (−0.111 + 0.433i)15-s + 0.250·16-s − 0.911·17-s + (0.341 − 0.619i)18-s + 0.617i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.568 - 0.822i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.568 - 0.822i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1470\)    =    \(2 \cdot 3 \cdot 5 \cdot 7^{2}\)
Sign: $-0.568 - 0.822i$
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: \((2,\ 1470,\ (\ :1/2),\ -0.568 - 0.822i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7247924916\)
\(L(\frac12)\) \(\approx\) \(0.7247924916\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.431 + 1.67i)T \)
5 \( 1 + T \)
7 \( 1 \)
good11 \( 1 - 5.13iT - 11T^{2} \)
13 \( 1 + 5.00iT - 13T^{2} \)
17 \( 1 + 3.75T + 17T^{2} \)
19 \( 1 - 2.69iT - 19T^{2} \)
23 \( 1 - 2.48iT - 23T^{2} \)
29 \( 1 - 6.18iT - 29T^{2} \)
31 \( 1 - 4.77iT - 31T^{2} \)
37 \( 1 - 0.0524T + 37T^{2} \)
41 \( 1 - 6.55T + 41T^{2} \)
43 \( 1 + 9.45T + 43T^{2} \)
47 \( 1 + 3.06T + 47T^{2} \)
53 \( 1 - 1.11iT - 53T^{2} \)
59 \( 1 - 13.2T + 59T^{2} \)
61 \( 1 - 4.14iT - 61T^{2} \)
67 \( 1 + 0.896T + 67T^{2} \)
71 \( 1 - 13.4iT - 71T^{2} \)
73 \( 1 - 6.98iT - 73T^{2} \)
79 \( 1 + 16.7T + 79T^{2} \)
83 \( 1 + 1.37T + 83T^{2} \)
89 \( 1 + 5.34T + 89T^{2} \)
97 \( 1 - 0.633iT - 97T^{2} \)
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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

−9.633039340958211805289563336417, −8.595885306306760511696057620000, −8.121708208852282417748379868084, −7.13999640406178074636146277361, −6.97573153492985533859466699619, −5.76221822770413095003943629042, −4.98482289985059110605960163231, −3.83994954126886145486242112605, −2.72269968830714293952660651531, −1.39373925773196660159062661108, 0.28480156149793925470495494164, 2.21103038615081307720873244817, 3.17475525787031952259295541263, 4.09805791674907264687737833790, 4.61445918620360335602500222916, 5.77161782540050918849315897426, 6.68624461146958251731185500505, 8.045132501304715012984509973468, 8.672619583490945951892892589156, 9.230754426363298746113075117588

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