Properties

Label 2-8470-1.1-c1-0-153
Degree $2$
Conductor $8470$
Sign $1$
Analytic cond. $67.6332$
Root an. cond. $8.22394$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 2.81·3-s + 4-s + 5-s + 2.81·6-s + 7-s + 8-s + 4.91·9-s + 10-s + 2.81·12-s + 0.500·13-s + 14-s + 2.81·15-s + 16-s − 2.39·17-s + 4.91·18-s − 3.43·19-s + 20-s + 2.81·21-s + 3.60·23-s + 2.81·24-s + 25-s + 0.500·26-s + 5.39·27-s + 28-s + 4.27·29-s + 2.81·30-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.62·3-s + 0.5·4-s + 0.447·5-s + 1.14·6-s + 0.377·7-s + 0.353·8-s + 1.63·9-s + 0.316·10-s + 0.812·12-s + 0.138·13-s + 0.267·14-s + 0.726·15-s + 0.250·16-s − 0.580·17-s + 1.15·18-s − 0.786·19-s + 0.223·20-s + 0.613·21-s + 0.751·23-s + 0.574·24-s + 0.200·25-s + 0.0982·26-s + 1.03·27-s + 0.188·28-s + 0.793·29-s + 0.513·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8470\)    =    \(2 \cdot 5 \cdot 7 \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(67.6332\)
Root analytic conductor: \(8.22394\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{8470} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 8470,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(7.451524636\)
\(L(\frac12)\) \(\approx\) \(7.451524636\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
5 \( 1 - T \)
7 \( 1 - T \)
11 \( 1 \)
good3 \( 1 - 2.81T + 3T^{2} \)
13 \( 1 - 0.500T + 13T^{2} \)
17 \( 1 + 2.39T + 17T^{2} \)
19 \( 1 + 3.43T + 19T^{2} \)
23 \( 1 - 3.60T + 23T^{2} \)
29 \( 1 - 4.27T + 29T^{2} \)
31 \( 1 - 3.10T + 31T^{2} \)
37 \( 1 + 2.27T + 37T^{2} \)
41 \( 1 - 0.0899T + 41T^{2} \)
43 \( 1 + 3.19T + 43T^{2} \)
47 \( 1 + 1.29T + 47T^{2} \)
53 \( 1 - 8.60T + 53T^{2} \)
59 \( 1 - 5.89T + 59T^{2} \)
61 \( 1 - 9.25T + 61T^{2} \)
67 \( 1 - 3.90T + 67T^{2} \)
71 \( 1 - 15.5T + 71T^{2} \)
73 \( 1 - 7.66T + 73T^{2} \)
79 \( 1 + 10.3T + 79T^{2} \)
83 \( 1 - 4.19T + 83T^{2} \)
89 \( 1 + 16.2T + 89T^{2} \)
97 \( 1 - 15.9T + 97T^{2} \)
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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

−7.978958364386973607666760342664, −6.88825640947390674279773752815, −6.72081530983282189768655119804, −5.58271753260597078533545943512, −4.83543308027246360708306495146, −4.11547546350474263545312055601, −3.48373270895975164889535566770, −2.56077824661636780836285982655, −2.21564991979225514355121374008, −1.20074143329364804525213671086, 1.20074143329364804525213671086, 2.21564991979225514355121374008, 2.56077824661636780836285982655, 3.48373270895975164889535566770, 4.11547546350474263545312055601, 4.83543308027246360708306495146, 5.58271753260597078533545943512, 6.72081530983282189768655119804, 6.88825640947390674279773752815, 7.978958364386973607666760342664

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