Properties

Label 2-8470-1.1-c1-0-115
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 − 3-s + 4-s + 5-s + 6-s − 7-s − 8-s − 2·9-s − 10-s − 12-s − 4.46·13-s + 14-s − 15-s + 16-s + 7.46·17-s + 2·18-s − 5.92·19-s + 20-s + 21-s − 4.46·23-s + 24-s + 25-s + 4.46·26-s + 5·27-s − 28-s + 6.92·29-s + 30-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.447·5-s + 0.408·6-s − 0.377·7-s − 0.353·8-s − 0.666·9-s − 0.316·10-s − 0.288·12-s − 1.23·13-s + 0.267·14-s − 0.258·15-s + 0.250·16-s + 1.81·17-s + 0.471·18-s − 1.36·19-s + 0.223·20-s + 0.218·21-s − 0.930·23-s + 0.204·24-s + 0.200·25-s + 0.875·26-s + 0.962·27-s − 0.188·28-s + 1.28·29-s + 0.182·30-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 \( 1 + T + 3T^{2} \)
13 \( 1 + 4.46T + 13T^{2} \)
17 \( 1 - 7.46T + 17T^{2} \)
19 \( 1 + 5.92T + 19T^{2} \)
23 \( 1 + 4.46T + 23T^{2} \)
29 \( 1 - 6.92T + 29T^{2} \)
31 \( 1 + 0.535T + 31T^{2} \)
37 \( 1 + 8.92T + 37T^{2} \)
41 \( 1 - 8.39T + 41T^{2} \)
43 \( 1 - 12.3T + 43T^{2} \)
47 \( 1 + 0.535T + 47T^{2} \)
53 \( 1 + 7.46T + 53T^{2} \)
59 \( 1 - 9.92T + 59T^{2} \)
61 \( 1 + 4.92T + 61T^{2} \)
67 \( 1 - 6.39T + 67T^{2} \)
71 \( 1 - 8T + 71T^{2} \)
73 \( 1 - 0.535T + 73T^{2} \)
79 \( 1 - 7.53T + 79T^{2} \)
83 \( 1 - T + 83T^{2} \)
89 \( 1 + 17.4T + 89T^{2} \)
97 \( 1 + 1.07T + 97T^{2} \)
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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

−7.50901428243533981388364086178, −6.70607673978891963069265732972, −6.10241713976649208912922375505, −5.55988060780869967644727257781, −4.85299517677108427706779932115, −3.80327169940936606400986485839, −2.78995075912620420923943364598, −2.23489239307226375874739187761, −0.994718781229834995507983955976, 0, 0.994718781229834995507983955976, 2.23489239307226375874739187761, 2.78995075912620420923943364598, 3.80327169940936606400986485839, 4.85299517677108427706779932115, 5.55988060780869967644727257781, 6.10241713976649208912922375505, 6.70607673978891963069265732972, 7.50901428243533981388364086178

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