Properties

Label 2-8470-1.1-c1-0-73
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 2.04·3-s + 4-s − 5-s + 2.04·6-s − 7-s − 8-s + 1.18·9-s + 10-s − 2.04·12-s − 3.50·13-s + 14-s + 2.04·15-s + 16-s + 4.28·17-s − 1.18·18-s − 4.44·19-s − 20-s + 2.04·21-s − 2·23-s + 2.04·24-s + 25-s + 3.50·26-s + 3.70·27-s − 28-s − 1.30·29-s − 2.04·30-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.18·3-s + 0.5·4-s − 0.447·5-s + 0.835·6-s − 0.377·7-s − 0.353·8-s + 0.396·9-s + 0.316·10-s − 0.590·12-s − 0.971·13-s + 0.267·14-s + 0.528·15-s + 0.250·16-s + 1.03·17-s − 0.280·18-s − 1.02·19-s − 0.223·20-s + 0.446·21-s − 0.417·23-s + 0.417·24-s + 0.200·25-s + 0.686·26-s + 0.713·27-s − 0.188·28-s − 0.242·29-s − 0.373·30-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: \(1\)
Selberg data: \((2,\ 8470,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.04T + 3T^{2} \)
13 \( 1 + 3.50T + 13T^{2} \)
17 \( 1 - 4.28T + 17T^{2} \)
19 \( 1 + 4.44T + 19T^{2} \)
23 \( 1 + 2T + 23T^{2} \)
29 \( 1 + 1.30T + 29T^{2} \)
31 \( 1 + 5.70T + 31T^{2} \)
37 \( 1 + 0.327T + 37T^{2} \)
41 \( 1 - 3.87T + 41T^{2} \)
43 \( 1 - 2.63T + 43T^{2} \)
47 \( 1 - 12.2T + 47T^{2} \)
53 \( 1 - 1.09T + 53T^{2} \)
59 \( 1 - 10.0T + 59T^{2} \)
61 \( 1 + 13.0T + 61T^{2} \)
67 \( 1 + 8.44T + 67T^{2} \)
71 \( 1 + 5.87T + 71T^{2} \)
73 \( 1 - 7.44T + 73T^{2} \)
79 \( 1 + 0.498T + 79T^{2} \)
83 \( 1 - 12.6T + 83T^{2} \)
89 \( 1 - 1.39T + 89T^{2} \)
97 \( 1 + 9.79T + 97T^{2} \)
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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

−7.36402212856831491653661156963, −6.89386399912371742066855801629, −5.95646203867408596420462089869, −5.66373512038698636977936137286, −4.73545005867717265065072720047, −3.95122225716364527548652146819, −2.97376422352147085524106343460, −2.05301745098409672903058895852, −0.811222802022697145909076791419, 0, 0.811222802022697145909076791419, 2.05301745098409672903058895852, 2.97376422352147085524106343460, 3.95122225716364527548652146819, 4.73545005867717265065072720047, 5.66373512038698636977936137286, 5.95646203867408596420462089869, 6.89386399912371742066855801629, 7.36402212856831491653661156963

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