Properties

Label 2-8470-1.1-c1-0-203
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 + 1.61·3-s + 4-s − 5-s + 1.61·6-s − 7-s + 8-s − 0.381·9-s − 10-s + 1.61·12-s + 5.23·13-s − 14-s − 1.61·15-s + 16-s − 2.61·17-s − 0.381·18-s − 7.85·19-s − 20-s − 1.61·21-s − 0.472·23-s + 1.61·24-s + 25-s + 5.23·26-s − 5.47·27-s − 28-s − 4.47·29-s − 1.61·30-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.934·3-s + 0.5·4-s − 0.447·5-s + 0.660·6-s − 0.377·7-s + 0.353·8-s − 0.127·9-s − 0.316·10-s + 0.467·12-s + 1.45·13-s − 0.267·14-s − 0.417·15-s + 0.250·16-s − 0.634·17-s − 0.0900·18-s − 1.80·19-s − 0.223·20-s − 0.353·21-s − 0.0984·23-s + 0.330·24-s + 0.200·25-s + 1.02·26-s − 1.05·27-s − 0.188·28-s − 0.830·29-s − 0.295·30-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 \( 1 - 1.61T + 3T^{2} \)
13 \( 1 - 5.23T + 13T^{2} \)
17 \( 1 + 2.61T + 17T^{2} \)
19 \( 1 + 7.85T + 19T^{2} \)
23 \( 1 + 0.472T + 23T^{2} \)
29 \( 1 + 4.47T + 29T^{2} \)
31 \( 1 - 2.47T + 31T^{2} \)
37 \( 1 + 11.2T + 37T^{2} \)
41 \( 1 + 2.85T + 41T^{2} \)
43 \( 1 - 2.09T + 43T^{2} \)
47 \( 1 - 4.76T + 47T^{2} \)
53 \( 1 + 2.76T + 53T^{2} \)
59 \( 1 - 4.14T + 59T^{2} \)
61 \( 1 + 13.7T + 61T^{2} \)
67 \( 1 + 3.38T + 67T^{2} \)
71 \( 1 - 10.9T + 71T^{2} \)
73 \( 1 + 2.85T + 73T^{2} \)
79 \( 1 + 4T + 79T^{2} \)
83 \( 1 - 8.85T + 83T^{2} \)
89 \( 1 + 14.8T + 89T^{2} \)
97 \( 1 + 8.38T + 97T^{2} \)
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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

−7.45034175310107062002090660040, −6.61527408571093802816398623923, −6.17344506689355132454695042873, −5.35998975045016349058030706891, −4.31757136168024228817471570209, −3.82549103373577736432633238536, −3.24936618577266879197907249320, −2.40087215492088515732067346828, −1.62478535356923383953808221363, 0, 1.62478535356923383953808221363, 2.40087215492088515732067346828, 3.24936618577266879197907249320, 3.82549103373577736432633238536, 4.31757136168024228817471570209, 5.35998975045016349058030706891, 6.17344506689355132454695042873, 6.61527408571093802816398623923, 7.45034175310107062002090660040

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