Properties

Label 2-8470-1.1-c1-0-74
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 − 1.34·3-s + 4-s + 5-s − 1.34·6-s + 7-s + 8-s − 1.19·9-s + 10-s − 1.34·12-s + 4.31·13-s + 14-s − 1.34·15-s + 16-s − 2.63·17-s − 1.19·18-s + 1.56·19-s + 20-s − 1.34·21-s + 5.45·23-s − 1.34·24-s + 25-s + 4.31·26-s + 5.63·27-s + 28-s + 2.83·29-s − 1.34·30-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.775·3-s + 0.5·4-s + 0.447·5-s − 0.548·6-s + 0.377·7-s + 0.353·8-s − 0.398·9-s + 0.316·10-s − 0.387·12-s + 1.19·13-s + 0.267·14-s − 0.346·15-s + 0.250·16-s − 0.639·17-s − 0.282·18-s + 0.359·19-s + 0.223·20-s − 0.293·21-s + 1.13·23-s − 0.274·24-s + 0.200·25-s + 0.845·26-s + 1.08·27-s + 0.188·28-s + 0.527·29-s − 0.245·30-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

\(L(1)\) \(\approx\) \(3.000554379\)
\(L(\frac12)\) \(\approx\) \(3.000554379\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.34T + 3T^{2} \)
13 \( 1 - 4.31T + 13T^{2} \)
17 \( 1 + 2.63T + 17T^{2} \)
19 \( 1 - 1.56T + 19T^{2} \)
23 \( 1 - 5.45T + 23T^{2} \)
29 \( 1 - 2.83T + 29T^{2} \)
31 \( 1 - 1.14T + 31T^{2} \)
37 \( 1 + 0.839T + 37T^{2} \)
41 \( 1 - 3.47T + 41T^{2} \)
43 \( 1 + 2.01T + 43T^{2} \)
47 \( 1 + 8.39T + 47T^{2} \)
53 \( 1 - 1.40T + 53T^{2} \)
59 \( 1 - 2.32T + 59T^{2} \)
61 \( 1 + 5.24T + 61T^{2} \)
67 \( 1 + 1.60T + 67T^{2} \)
71 \( 1 + 13.8T + 71T^{2} \)
73 \( 1 + 1.14T + 73T^{2} \)
79 \( 1 - 9.65T + 79T^{2} \)
83 \( 1 + 2.51T + 83T^{2} \)
89 \( 1 - 2.19T + 89T^{2} \)
97 \( 1 + 7.56T + 97T^{2} \)
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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

−7.61512324741244772217488290096, −6.80322001226660149345022720374, −6.22185267666527319959451103168, −5.77294609610918704339465355540, −4.97715718474848267954870814228, −4.55647032240167897051495014616, −3.46997565712715089435385083093, −2.80708026547922481827297384406, −1.74142416255422947762213437232, −0.821235043728869545470673534759, 0.821235043728869545470673534759, 1.74142416255422947762213437232, 2.80708026547922481827297384406, 3.46997565712715089435385083093, 4.55647032240167897051495014616, 4.97715718474848267954870814228, 5.77294609610918704339465355540, 6.22185267666527319959451103168, 6.80322001226660149345022720374, 7.61512324741244772217488290096

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