Properties

Label 2-8470-1.1-c1-0-201
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 + 0.414·3-s + 4-s + 5-s + 0.414·6-s − 7-s + 8-s − 2.82·9-s + 10-s + 0.414·12-s + 2.13·13-s − 14-s + 0.414·15-s + 16-s + 4.44·17-s − 2.82·18-s − 8.27·19-s + 20-s − 0.414·21-s − 0.646·23-s + 0.414·24-s + 25-s + 2.13·26-s − 2.41·27-s − 28-s + 0.428·29-s + 0.414·30-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.239·3-s + 0.5·4-s + 0.447·5-s + 0.169·6-s − 0.377·7-s + 0.353·8-s − 0.942·9-s + 0.316·10-s + 0.119·12-s + 0.591·13-s − 0.267·14-s + 0.106·15-s + 0.250·16-s + 1.07·17-s − 0.666·18-s − 1.89·19-s + 0.223·20-s − 0.0903·21-s − 0.134·23-s + 0.0845·24-s + 0.200·25-s + 0.418·26-s − 0.464·27-s − 0.188·28-s + 0.0796·29-s + 0.0756·30-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 \( 1 - 0.414T + 3T^{2} \)
13 \( 1 - 2.13T + 13T^{2} \)
17 \( 1 - 4.44T + 17T^{2} \)
19 \( 1 + 8.27T + 19T^{2} \)
23 \( 1 + 0.646T + 23T^{2} \)
29 \( 1 - 0.428T + 29T^{2} \)
31 \( 1 + 4.52T + 31T^{2} \)
37 \( 1 + 8.49T + 37T^{2} \)
41 \( 1 + 10.6T + 41T^{2} \)
43 \( 1 + 2.65T + 43T^{2} \)
47 \( 1 - 4.34T + 47T^{2} \)
53 \( 1 - 0.0206T + 53T^{2} \)
59 \( 1 + 4.51T + 59T^{2} \)
61 \( 1 + 1.36T + 61T^{2} \)
67 \( 1 + 2.68T + 67T^{2} \)
71 \( 1 + 9.05T + 71T^{2} \)
73 \( 1 - 0.585T + 73T^{2} \)
79 \( 1 + 4.24T + 79T^{2} \)
83 \( 1 - 4.62T + 83T^{2} \)
89 \( 1 + 2.17T + 89T^{2} \)
97 \( 1 - 13.0T + 97T^{2} \)
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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

−7.29943224893690894400538899764, −6.55870811316111722318936123646, −5.98953364147231876381699403842, −5.47579644128930334928571994644, −4.67466461003413361608446759902, −3.67994147396936548168109004166, −3.25550862994466498584217813827, −2.32549793410702565266513341996, −1.56624819212278216523012407055, 0, 1.56624819212278216523012407055, 2.32549793410702565266513341996, 3.25550862994466498584217813827, 3.67994147396936548168109004166, 4.67466461003413361608446759902, 5.47579644128930334928571994644, 5.98953364147231876381699403842, 6.55870811316111722318936123646, 7.29943224893690894400538899764

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