Properties

Label 2-8470-1.1-c1-0-41
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 0.365·3-s + 4-s − 5-s + 0.365·6-s − 7-s + 8-s − 2.86·9-s − 10-s + 0.365·12-s − 2.20·13-s − 14-s − 0.365·15-s + 16-s + 6.26·17-s − 2.86·18-s − 6.00·19-s − 20-s − 0.365·21-s + 0.452·23-s + 0.365·24-s + 25-s − 2.20·26-s − 2.14·27-s − 28-s − 1.90·29-s − 0.365·30-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.211·3-s + 0.5·4-s − 0.447·5-s + 0.149·6-s − 0.377·7-s + 0.353·8-s − 0.955·9-s − 0.316·10-s + 0.105·12-s − 0.610·13-s − 0.267·14-s − 0.0944·15-s + 0.250·16-s + 1.51·17-s − 0.675·18-s − 1.37·19-s − 0.223·20-s − 0.0798·21-s + 0.0942·23-s + 0.0746·24-s + 0.200·25-s − 0.431·26-s − 0.412·27-s − 0.188·28-s − 0.354·29-s − 0.0667·30-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: \(0\)
Selberg data: \((2,\ 8470,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.385145833\)
\(L(\frac12)\) \(\approx\) \(2.385145833\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.365T + 3T^{2} \)
13 \( 1 + 2.20T + 13T^{2} \)
17 \( 1 - 6.26T + 17T^{2} \)
19 \( 1 + 6.00T + 19T^{2} \)
23 \( 1 - 0.452T + 23T^{2} \)
29 \( 1 + 1.90T + 29T^{2} \)
31 \( 1 + 6.05T + 31T^{2} \)
37 \( 1 - 3.54T + 37T^{2} \)
41 \( 1 + 0.0206T + 41T^{2} \)
43 \( 1 - 3.71T + 43T^{2} \)
47 \( 1 - 8.63T + 47T^{2} \)
53 \( 1 - 8.03T + 53T^{2} \)
59 \( 1 + 12.7T + 59T^{2} \)
61 \( 1 - 6.02T + 61T^{2} \)
67 \( 1 - 8.19T + 67T^{2} \)
71 \( 1 - 15.7T + 71T^{2} \)
73 \( 1 - 12.7T + 73T^{2} \)
79 \( 1 - 10.7T + 79T^{2} \)
83 \( 1 - 10.9T + 83T^{2} \)
89 \( 1 + 9.56T + 89T^{2} \)
97 \( 1 + 4.22T + 97T^{2} \)
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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

−7.82354812038824019837928926831, −7.04787122153371021453412512658, −6.33807632679231883576620967660, −5.58506840329920456555821992968, −5.13871189717080976082794593205, −4.04808341264701195774809434050, −3.60931045069813327689835990615, −2.75112978452970531929764305336, −2.10418885573954133631208542428, −0.64115784489123138212963027147, 0.64115784489123138212963027147, 2.10418885573954133631208542428, 2.75112978452970531929764305336, 3.60931045069813327689835990615, 4.04808341264701195774809434050, 5.13871189717080976082794593205, 5.58506840329920456555821992968, 6.33807632679231883576620967660, 7.04787122153371021453412512658, 7.82354812038824019837928926831

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