Properties

Label 2-8470-1.1-c1-0-55
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 + 0.289·3-s + 4-s − 5-s + 0.289·6-s + 7-s + 8-s − 2.91·9-s − 10-s + 0.289·12-s + 0.289·13-s + 14-s − 0.289·15-s + 16-s − 0.916·17-s − 2.91·18-s − 3.20·19-s − 20-s + 0.289·21-s − 5.33·23-s + 0.289·24-s + 25-s + 0.289·26-s − 1.71·27-s + 28-s + 9.83·29-s − 0.289·30-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.166·3-s + 0.5·4-s − 0.447·5-s + 0.118·6-s + 0.377·7-s + 0.353·8-s − 0.972·9-s − 0.316·10-s + 0.0834·12-s + 0.0802·13-s + 0.267·14-s − 0.0746·15-s + 0.250·16-s − 0.222·17-s − 0.687·18-s − 0.735·19-s − 0.223·20-s + 0.0631·21-s − 1.11·23-s + 0.0590·24-s + 0.200·25-s + 0.0567·26-s − 0.329·27-s + 0.188·28-s + 1.82·29-s − 0.0527·30-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

\(L(1)\) \(\approx\) \(2.799757082\)
\(L(\frac12)\) \(\approx\) \(2.799757082\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.289T + 3T^{2} \)
13 \( 1 - 0.289T + 13T^{2} \)
17 \( 1 + 0.916T + 17T^{2} \)
19 \( 1 + 3.20T + 19T^{2} \)
23 \( 1 + 5.33T + 23T^{2} \)
29 \( 1 - 9.83T + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + 9.25T + 37T^{2} \)
41 \( 1 + 0.578T + 41T^{2} \)
43 \( 1 - 5.75T + 43T^{2} \)
47 \( 1 - 11.2T + 47T^{2} \)
53 \( 1 - 4.91T + 53T^{2} \)
59 \( 1 - 7.20T + 59T^{2} \)
61 \( 1 - 9.49T + 61T^{2} \)
67 \( 1 - 10.7T + 67T^{2} \)
71 \( 1 - 1.66T + 71T^{2} \)
73 \( 1 - 10.3T + 73T^{2} \)
79 \( 1 + 2.42T + 79T^{2} \)
83 \( 1 - 2.28T + 83T^{2} \)
89 \( 1 - 3.42T + 89T^{2} \)
97 \( 1 + 6.33T + 97T^{2} \)
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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

−7.82317464091137713829717402126, −6.97527323139160506707512202520, −6.38067569494574578714195185231, −5.61545770335549493674546235921, −5.01914665972444400290957349314, −4.14598666212608247994587603083, −3.66333959878010799718416476833, −2.64440171493433481428624307734, −2.10459948107619394671808123086, −0.70622398268489978735753856046, 0.70622398268489978735753856046, 2.10459948107619394671808123086, 2.64440171493433481428624307734, 3.66333959878010799718416476833, 4.14598666212608247994587603083, 5.01914665972444400290957349314, 5.61545770335549493674546235921, 6.38067569494574578714195185231, 6.97527323139160506707512202520, 7.82317464091137713829717402126

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