Properties

Label 2-8470-1.1-c1-0-147
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 + 3.18·3-s + 4-s − 5-s + 3.18·6-s − 7-s + 8-s + 7.16·9-s − 10-s + 3.18·12-s + 5.69·13-s − 14-s − 3.18·15-s + 16-s − 0.749·17-s + 7.16·18-s + 6.78·19-s − 20-s − 3.18·21-s + 3.94·23-s + 3.18·24-s + 25-s + 5.69·26-s + 13.2·27-s − 28-s − 5.73·29-s − 3.18·30-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.84·3-s + 0.5·4-s − 0.447·5-s + 1.30·6-s − 0.377·7-s + 0.353·8-s + 2.38·9-s − 0.316·10-s + 0.920·12-s + 1.57·13-s − 0.267·14-s − 0.823·15-s + 0.250·16-s − 0.181·17-s + 1.68·18-s + 1.55·19-s − 0.223·20-s − 0.695·21-s + 0.821·23-s + 0.650·24-s + 0.200·25-s + 1.11·26-s + 2.55·27-s − 0.188·28-s − 1.06·29-s − 0.581·30-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

\(L(1)\) \(\approx\) \(7.072441461\)
\(L(\frac12)\) \(\approx\) \(7.072441461\)
\(L(\frac{3}{2})\) 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 \( 1 - 3.18T + 3T^{2} \)
13 \( 1 - 5.69T + 13T^{2} \)
17 \( 1 + 0.749T + 17T^{2} \)
19 \( 1 - 6.78T + 19T^{2} \)
23 \( 1 - 3.94T + 23T^{2} \)
29 \( 1 + 5.73T + 29T^{2} \)
31 \( 1 + 7.10T + 31T^{2} \)
37 \( 1 + 11.6T + 37T^{2} \)
41 \( 1 - 11.4T + 41T^{2} \)
43 \( 1 - 2.33T + 43T^{2} \)
47 \( 1 + 5.00T + 47T^{2} \)
53 \( 1 + 10.7T + 53T^{2} \)
59 \( 1 - 5.22T + 59T^{2} \)
61 \( 1 - 7.68T + 61T^{2} \)
67 \( 1 + 7.39T + 67T^{2} \)
71 \( 1 + 0.575T + 71T^{2} \)
73 \( 1 + 12.5T + 73T^{2} \)
79 \( 1 - 6.83T + 79T^{2} \)
83 \( 1 - 10.5T + 83T^{2} \)
89 \( 1 - 4.80T + 89T^{2} \)
97 \( 1 - 16.2T + 97T^{2} \)
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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

−7.55965869144701947409529311549, −7.41220720340725167561629154586, −6.55369310984045896775163737649, −5.65539605283349356356340380832, −4.79486638996496190547738463558, −3.79434862675326071176715976063, −3.53065380777963496499476912937, −3.01394940324880800209783699245, −1.98494038726488819603554362848, −1.17615636695872939995265826848, 1.17615636695872939995265826848, 1.98494038726488819603554362848, 3.01394940324880800209783699245, 3.53065380777963496499476912937, 3.79434862675326071176715976063, 4.79486638996496190547738463558, 5.65539605283349356356340380832, 6.55369310984045896775163737649, 7.41220720340725167561629154586, 7.55965869144701947409529311549

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