Properties

Label 2-8470-1.1-c1-0-192
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 + 2.23·3-s + 4-s − 5-s − 2.23·6-s + 7-s − 8-s + 2.00·9-s + 10-s + 2.23·12-s + 13-s − 14-s − 2.23·15-s + 16-s + 3.23·17-s − 2.00·18-s + 2.23·19-s − 20-s + 2.23·21-s − 8.23·23-s − 2.23·24-s + 25-s − 26-s − 2.23·27-s + 28-s − 4.47·29-s + 2.23·30-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.29·3-s + 0.5·4-s − 0.447·5-s − 0.912·6-s + 0.377·7-s − 0.353·8-s + 0.666·9-s + 0.316·10-s + 0.645·12-s + 0.277·13-s − 0.267·14-s − 0.577·15-s + 0.250·16-s + 0.784·17-s − 0.471·18-s + 0.512·19-s − 0.223·20-s + 0.487·21-s − 1.71·23-s − 0.456·24-s + 0.200·25-s − 0.196·26-s − 0.430·27-s + 0.188·28-s − 0.830·29-s + 0.408·30-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: \(1\)
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 \( 1 - 2.23T + 3T^{2} \)
13 \( 1 - T + 13T^{2} \)
17 \( 1 - 3.23T + 17T^{2} \)
19 \( 1 - 2.23T + 19T^{2} \)
23 \( 1 + 8.23T + 23T^{2} \)
29 \( 1 + 4.47T + 29T^{2} \)
31 \( 1 + 0.763T + 31T^{2} \)
37 \( 1 + 8.94T + 37T^{2} \)
41 \( 1 + 2.76T + 41T^{2} \)
43 \( 1 + 1.23T + 43T^{2} \)
47 \( 1 + 5.70T + 47T^{2} \)
53 \( 1 - 7.70T + 53T^{2} \)
59 \( 1 + 8.70T + 59T^{2} \)
61 \( 1 + 12.4T + 61T^{2} \)
67 \( 1 + 2.76T + 67T^{2} \)
71 \( 1 + 10.4T + 71T^{2} \)
73 \( 1 + 2.76T + 73T^{2} \)
79 \( 1 - 5.76T + 79T^{2} \)
83 \( 1 - 4.23T + 83T^{2} \)
89 \( 1 + 12.7T + 89T^{2} \)
97 \( 1 - 12.4T + 97T^{2} \)
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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

−7.59678279233050086686461187014, −7.24700723622687158554385952442, −6.15271164089432478869951377365, −5.47059377605500960068677965096, −4.41164144425328976219731921180, −3.54391758655861140139800648002, −3.15105347525637893361597702738, −2.06557513668394933611581132643, −1.47945003216174396944052996806, 0, 1.47945003216174396944052996806, 2.06557513668394933611581132643, 3.15105347525637893361597702738, 3.54391758655861140139800648002, 4.41164144425328976219731921180, 5.47059377605500960068677965096, 6.15271164089432478869951377365, 7.24700723622687158554385952442, 7.59678279233050086686461187014

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