Properties

Label 2-8470-1.1-c1-0-15
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 − 3.14·3-s + 4-s + 5-s + 3.14·6-s − 7-s − 8-s + 6.91·9-s − 10-s − 3.14·12-s + 5.43·13-s + 14-s − 3.14·15-s + 16-s − 0.477·17-s − 6.91·18-s − 5.91·19-s + 20-s + 3.14·21-s − 2.65·23-s + 3.14·24-s + 25-s − 5.43·26-s − 12.3·27-s − 28-s − 6.56·29-s + 3.14·30-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.81·3-s + 0.5·4-s + 0.447·5-s + 1.28·6-s − 0.377·7-s − 0.353·8-s + 2.30·9-s − 0.316·10-s − 0.909·12-s + 1.50·13-s + 0.267·14-s − 0.813·15-s + 0.250·16-s − 0.115·17-s − 1.63·18-s − 1.35·19-s + 0.223·20-s + 0.687·21-s − 0.553·23-s + 0.642·24-s + 0.200·25-s − 1.06·26-s − 2.37·27-s − 0.188·28-s − 1.21·29-s + 0.574·30-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

\(L(1)\) \(\approx\) \(0.5525776631\)
\(L(\frac12)\) \(\approx\) \(0.5525776631\)
\(L(\frac{3}{2})\) 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 \( 1 + 3.14T + 3T^{2} \)
13 \( 1 - 5.43T + 13T^{2} \)
17 \( 1 + 0.477T + 17T^{2} \)
19 \( 1 + 5.91T + 19T^{2} \)
23 \( 1 + 2.65T + 23T^{2} \)
29 \( 1 + 6.56T + 29T^{2} \)
31 \( 1 + 9.68T + 31T^{2} \)
37 \( 1 + 4.56T + 37T^{2} \)
41 \( 1 - 2.52T + 41T^{2} \)
43 \( 1 - 11.9T + 43T^{2} \)
47 \( 1 - 9.42T + 47T^{2} \)
53 \( 1 - 6.15T + 53T^{2} \)
59 \( 1 + 0.566T + 59T^{2} \)
61 \( 1 - 2.04T + 61T^{2} \)
67 \( 1 + 1.24T + 67T^{2} \)
71 \( 1 + 3.62T + 71T^{2} \)
73 \( 1 - 15.6T + 73T^{2} \)
79 \( 1 + 3.38T + 79T^{2} \)
83 \( 1 - 1.73T + 83T^{2} \)
89 \( 1 + 9.54T + 89T^{2} \)
97 \( 1 + 3.93T + 97T^{2} \)
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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

−7.57298612175362493596211186114, −6.93856863293992979910574524579, −6.35960508074584708551202209746, −5.69478197349628451203653282658, −5.55460900138522389593334353634, −4.20603597283053100972634224128, −3.77225918159377119369412292607, −2.23221532799221848469404404589, −1.44722134034854773190647822756, −0.47133589874212518218099991935, 0.47133589874212518218099991935, 1.44722134034854773190647822756, 2.23221532799221848469404404589, 3.77225918159377119369412292607, 4.20603597283053100972634224128, 5.55460900138522389593334353634, 5.69478197349628451203653282658, 6.35960508074584708551202209746, 6.93856863293992979910574524579, 7.57298612175362493596211186114

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