Properties

Label 2-8470-1.1-c1-0-70
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 + 1.16·3-s + 4-s − 5-s + 1.16·6-s − 7-s + 8-s − 1.63·9-s − 10-s + 1.16·12-s − 1.43·13-s − 14-s − 1.16·15-s + 16-s + 7.28·17-s − 1.63·18-s + 4.33·19-s − 20-s − 1.16·21-s + 2.02·23-s + 1.16·24-s + 25-s − 1.43·26-s − 5.41·27-s − 28-s − 9.11·29-s − 1.16·30-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.675·3-s + 0.5·4-s − 0.447·5-s + 0.477·6-s − 0.377·7-s + 0.353·8-s − 0.543·9-s − 0.316·10-s + 0.337·12-s − 0.397·13-s − 0.267·14-s − 0.302·15-s + 0.250·16-s + 1.76·17-s − 0.384·18-s + 0.994·19-s − 0.223·20-s − 0.255·21-s + 0.422·23-s + 0.238·24-s + 0.200·25-s − 0.281·26-s − 1.04·27-s − 0.188·28-s − 1.69·29-s − 0.213·30-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 8470,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.601611596\)
\(L(\frac12)\) \(\approx\) \(3.601611596\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.16T + 3T^{2} \)
13 \( 1 + 1.43T + 13T^{2} \)
17 \( 1 - 7.28T + 17T^{2} \)
19 \( 1 - 4.33T + 19T^{2} \)
23 \( 1 - 2.02T + 23T^{2} \)
29 \( 1 + 9.11T + 29T^{2} \)
31 \( 1 - 3.65T + 31T^{2} \)
37 \( 1 + 3.11T + 37T^{2} \)
41 \( 1 - 2.05T + 41T^{2} \)
43 \( 1 - 0.705T + 43T^{2} \)
47 \( 1 - 4.00T + 47T^{2} \)
53 \( 1 + 4.87T + 53T^{2} \)
59 \( 1 - 12.3T + 59T^{2} \)
61 \( 1 - 6.62T + 61T^{2} \)
67 \( 1 + 1.89T + 67T^{2} \)
71 \( 1 - 0.421T + 71T^{2} \)
73 \( 1 - 7.37T + 73T^{2} \)
79 \( 1 - 3.98T + 79T^{2} \)
83 \( 1 + 7.15T + 83T^{2} \)
89 \( 1 - 8.34T + 89T^{2} \)
97 \( 1 - 11.3T + 97T^{2} \)
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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

−7.57238436701965892522915601698, −7.35898952157588210379057685227, −6.33209447558201850984785323660, −5.49299156494887822423506556381, −5.18894779807275860266654517027, −4.00427349200044265507741558680, −3.41272037817581324742117171200, −2.96892400884381755039938853448, −2.04206739253327604492646005995, −0.798532337890934416239650085139, 0.798532337890934416239650085139, 2.04206739253327604492646005995, 2.96892400884381755039938853448, 3.41272037817581324742117171200, 4.00427349200044265507741558680, 5.18894779807275860266654517027, 5.49299156494887822423506556381, 6.33209447558201850984785323660, 7.35898952157588210379057685227, 7.57238436701965892522915601698

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