Properties

Label 2-8470-1.1-c1-0-216
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 + 1.84·3-s + 4-s + 5-s + 1.84·6-s − 7-s + 8-s + 0.392·9-s + 10-s + 1.84·12-s − 1.78·13-s − 14-s + 1.84·15-s + 16-s − 7.29·17-s + 0.392·18-s − 3.17·19-s + 20-s − 1.84·21-s − 4.72·23-s + 1.84·24-s + 25-s − 1.78·26-s − 4.80·27-s − 28-s + 4.51·29-s + 1.84·30-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.06·3-s + 0.5·4-s + 0.447·5-s + 0.751·6-s − 0.377·7-s + 0.353·8-s + 0.130·9-s + 0.316·10-s + 0.531·12-s − 0.494·13-s − 0.267·14-s + 0.475·15-s + 0.250·16-s − 1.76·17-s + 0.0924·18-s − 0.727·19-s + 0.223·20-s − 0.401·21-s − 0.985·23-s + 0.375·24-s + 0.200·25-s − 0.349·26-s − 0.924·27-s − 0.188·28-s + 0.838·29-s + 0.336·30-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 \( 1 - 1.84T + 3T^{2} \)
13 \( 1 + 1.78T + 13T^{2} \)
17 \( 1 + 7.29T + 17T^{2} \)
19 \( 1 + 3.17T + 19T^{2} \)
23 \( 1 + 4.72T + 23T^{2} \)
29 \( 1 - 4.51T + 29T^{2} \)
31 \( 1 + 4.99T + 31T^{2} \)
37 \( 1 + 4.73T + 37T^{2} \)
41 \( 1 - 4.17T + 41T^{2} \)
43 \( 1 + 11.6T + 43T^{2} \)
47 \( 1 - 3.74T + 47T^{2} \)
53 \( 1 - 1.32T + 53T^{2} \)
59 \( 1 - 9.71T + 59T^{2} \)
61 \( 1 + 9.80T + 61T^{2} \)
67 \( 1 + 6.96T + 67T^{2} \)
71 \( 1 - 16.7T + 71T^{2} \)
73 \( 1 - 6.05T + 73T^{2} \)
79 \( 1 + 15.0T + 79T^{2} \)
83 \( 1 + 17.5T + 83T^{2} \)
89 \( 1 - 14.3T + 89T^{2} \)
97 \( 1 - 5.08T + 97T^{2} \)
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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

−7.31370607870860565429060604888, −6.72221128602981941593829212078, −6.11724105190863556269895873068, −5.31833625435808471805295376077, −4.46545907570758899360160493656, −3.86725054454041746865647200102, −3.02493676580132302758944729485, −2.31462741553372275049346561968, −1.84851509265254020088176189606, 0, 1.84851509265254020088176189606, 2.31462741553372275049346561968, 3.02493676580132302758944729485, 3.86725054454041746865647200102, 4.46545907570758899360160493656, 5.31833625435808471805295376077, 6.11724105190863556269895873068, 6.72221128602981941593829212078, 7.31370607870860565429060604888

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