Properties

Label 2-8470-1.1-c1-0-134
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 + 3.04·3-s + 4-s − 5-s + 3.04·6-s + 7-s + 8-s + 6.28·9-s − 10-s + 3.04·12-s + 0.353·13-s + 14-s − 3.04·15-s + 16-s + 0.810·17-s + 6.28·18-s − 2.59·19-s − 20-s + 3.04·21-s − 2·23-s + 3.04·24-s + 25-s + 0.353·26-s + 10.0·27-s + 28-s + 9.54·29-s − 3.04·30-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.75·3-s + 0.5·4-s − 0.447·5-s + 1.24·6-s + 0.377·7-s + 0.353·8-s + 2.09·9-s − 0.316·10-s + 0.879·12-s + 0.0979·13-s + 0.267·14-s − 0.786·15-s + 0.250·16-s + 0.196·17-s + 1.48·18-s − 0.594·19-s − 0.223·20-s + 0.664·21-s − 0.417·23-s + 0.621·24-s + 0.200·25-s + 0.0692·26-s + 1.92·27-s + 0.188·28-s + 1.77·29-s − 0.556·30-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

\(L(1)\) \(\approx\) \(6.821023696\)
\(L(\frac12)\) \(\approx\) \(6.821023696\)
\(L(\frac{3}{2})\) 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 \( 1 - 3.04T + 3T^{2} \)
13 \( 1 - 0.353T + 13T^{2} \)
17 \( 1 - 0.810T + 17T^{2} \)
19 \( 1 + 2.59T + 19T^{2} \)
23 \( 1 + 2T + 23T^{2} \)
29 \( 1 - 9.54T + 29T^{2} \)
31 \( 1 + 5.70T + 31T^{2} \)
37 \( 1 - 3.56T + 37T^{2} \)
41 \( 1 - 8.25T + 41T^{2} \)
43 \( 1 - 9.49T + 43T^{2} \)
47 \( 1 + 8.60T + 47T^{2} \)
53 \( 1 + 2.79T + 53T^{2} \)
59 \( 1 - 13.2T + 59T^{2} \)
61 \( 1 - 6.70T + 61T^{2} \)
67 \( 1 + 1.40T + 67T^{2} \)
71 \( 1 + 12.9T + 71T^{2} \)
73 \( 1 - 5.15T + 73T^{2} \)
79 \( 1 - 3.64T + 79T^{2} \)
83 \( 1 - 2.66T + 83T^{2} \)
89 \( 1 - 9.63T + 89T^{2} \)
97 \( 1 + 9.79T + 97T^{2} \)
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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

−8.005002540178920914796413223628, −7.20868561502510352030287720330, −6.58496310709772951487203593919, −5.64261027982406183786612124676, −4.62283714087448810873913752123, −4.14536815953383592872552742619, −3.51314921031663299894449343944, −2.70621969050571537283438950633, −2.17105705241084043025777071337, −1.12206128739584985024541740578, 1.12206128739584985024541740578, 2.17105705241084043025777071337, 2.70621969050571537283438950633, 3.51314921031663299894449343944, 4.14536815953383592872552742619, 4.62283714087448810873913752123, 5.64261027982406183786612124676, 6.58496310709772951487203593919, 7.20868561502510352030287720330, 8.005002540178920914796413223628

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