Properties

Label 2-8470-1.1-c1-0-127
Degree $2$
Conductor $8470$
Sign $-1$
Analytic cond. $67.6332$
Root an. cond. $8.22394$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 2·3-s + 4-s − 5-s − 2·6-s − 7-s + 8-s + 9-s − 10-s − 2·12-s − 2·13-s − 14-s + 2·15-s + 16-s + 6·17-s + 18-s − 2·19-s − 20-s + 2·21-s − 6·23-s − 2·24-s + 25-s − 2·26-s + 4·27-s − 28-s + 2·30-s + 8·31-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.15·3-s + 1/2·4-s − 0.447·5-s − 0.816·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s − 0.316·10-s − 0.577·12-s − 0.554·13-s − 0.267·14-s + 0.516·15-s + 1/4·16-s + 1.45·17-s + 0.235·18-s − 0.458·19-s − 0.223·20-s + 0.436·21-s − 1.25·23-s − 0.408·24-s + 1/5·25-s − 0.392·26-s + 0.769·27-s − 0.188·28-s + 0.365·30-s + 1.43·31-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: yes
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 \( 1 + 2 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 + 12 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 14 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 8 T + p T^{2} \)
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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

−7.19544417070016173435125149986, −6.60450088088082817599129972392, −5.91744933494826602852281952265, −5.43618965180923653974703994736, −4.74025588313786851554961465992, −4.00607382592463520702767907374, −3.24310861852235581590465707447, −2.36039041640977299742312702708, −1.09350189291722615835272882215, 0, 1.09350189291722615835272882215, 2.36039041640977299742312702708, 3.24310861852235581590465707447, 4.00607382592463520702767907374, 4.74025588313786851554961465992, 5.43618965180923653974703994736, 5.91744933494826602852281952265, 6.60450088088082817599129972392, 7.19544417070016173435125149986

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