Properties

Label 2-8470-1.1-c1-0-139
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 + 2.73·3-s + 4-s + 5-s + 2.73·6-s − 7-s + 8-s + 4.46·9-s + 10-s + 2.73·12-s + 1.46·13-s − 14-s + 2.73·15-s + 16-s − 3.46·17-s + 4.46·18-s + 2.73·19-s + 20-s − 2.73·21-s + 1.26·23-s + 2.73·24-s + 25-s + 1.46·26-s + 3.99·27-s − 28-s + 4.73·29-s + 2.73·30-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.57·3-s + 0.5·4-s + 0.447·5-s + 1.11·6-s − 0.377·7-s + 0.353·8-s + 1.48·9-s + 0.316·10-s + 0.788·12-s + 0.406·13-s − 0.267·14-s + 0.705·15-s + 0.250·16-s − 0.840·17-s + 1.05·18-s + 0.626·19-s + 0.223·20-s − 0.596·21-s + 0.264·23-s + 0.557·24-s + 0.200·25-s + 0.287·26-s + 0.769·27-s − 0.188·28-s + 0.878·29-s + 0.498·30-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

\(L(1)\) \(\approx\) \(6.920817964\)
\(L(\frac12)\) \(\approx\) \(6.920817964\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.73T + 3T^{2} \)
13 \( 1 - 1.46T + 13T^{2} \)
17 \( 1 + 3.46T + 17T^{2} \)
19 \( 1 - 2.73T + 19T^{2} \)
23 \( 1 - 1.26T + 23T^{2} \)
29 \( 1 - 4.73T + 29T^{2} \)
31 \( 1 - 8.92T + 31T^{2} \)
37 \( 1 - 3.26T + 37T^{2} \)
41 \( 1 - 4.73T + 41T^{2} \)
43 \( 1 - 4.92T + 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + 4.73T + 53T^{2} \)
59 \( 1 + 13.8T + 59T^{2} \)
61 \( 1 + 2T + 61T^{2} \)
67 \( 1 + 10.9T + 67T^{2} \)
71 \( 1 - 9.46T + 71T^{2} \)
73 \( 1 + 11.4T + 73T^{2} \)
79 \( 1 + 6.73T + 79T^{2} \)
83 \( 1 - 4.39T + 83T^{2} \)
89 \( 1 + 15.4T + 89T^{2} \)
97 \( 1 - 5.80T + 97T^{2} \)
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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

−7.78059940453279336224313384147, −7.15332615899911342776755347434, −6.36523581560699696948527877599, −5.85179520421783260878237406887, −4.63933161467128895793958257258, −4.29422184733729054405661842238, −3.19090855686503637223716267846, −2.89984070631018003729540512118, −2.11732139466118746846439438010, −1.14327275344601601510968457782, 1.14327275344601601510968457782, 2.11732139466118746846439438010, 2.89984070631018003729540512118, 3.19090855686503637223716267846, 4.29422184733729054405661842238, 4.63933161467128895793958257258, 5.85179520421783260878237406887, 6.36523581560699696948527877599, 7.15332615899911342776755347434, 7.78059940453279336224313384147

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