Properties

Label 2-8470-1.1-c1-0-99
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 + 1.91·3-s + 4-s − 5-s + 1.91·6-s + 7-s + 8-s + 0.683·9-s − 10-s + 1.91·12-s + 0.869·13-s + 14-s − 1.91·15-s + 16-s + 4.15·17-s + 0.683·18-s − 7.56·19-s − 20-s + 1.91·21-s − 2·23-s + 1.91·24-s + 25-s + 0.869·26-s − 4.44·27-s + 28-s + 1.19·29-s − 1.91·30-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.10·3-s + 0.5·4-s − 0.447·5-s + 0.783·6-s + 0.377·7-s + 0.353·8-s + 0.227·9-s − 0.316·10-s + 0.554·12-s + 0.241·13-s + 0.267·14-s − 0.495·15-s + 0.250·16-s + 1.00·17-s + 0.160·18-s − 1.73·19-s − 0.223·20-s + 0.418·21-s − 0.417·23-s + 0.391·24-s + 0.200·25-s + 0.170·26-s − 0.855·27-s + 0.188·28-s + 0.222·29-s − 0.350·30-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

\(L(1)\) \(\approx\) \(4.918667909\)
\(L(\frac12)\) \(\approx\) \(4.918667909\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.91T + 3T^{2} \)
13 \( 1 - 0.869T + 13T^{2} \)
17 \( 1 - 4.15T + 17T^{2} \)
19 \( 1 + 7.56T + 19T^{2} \)
23 \( 1 + 2T + 23T^{2} \)
29 \( 1 - 1.19T + 29T^{2} \)
31 \( 1 - 7.70T + 31T^{2} \)
37 \( 1 - 6.81T + 37T^{2} \)
41 \( 1 - 9.86T + 41T^{2} \)
43 \( 1 - 6.62T + 43T^{2} \)
47 \( 1 - 10.7T + 47T^{2} \)
53 \( 1 + 1.57T + 53T^{2} \)
59 \( 1 + 7.43T + 59T^{2} \)
61 \( 1 - 7.73T + 61T^{2} \)
67 \( 1 - 3.56T + 67T^{2} \)
71 \( 1 + 2.23T + 71T^{2} \)
73 \( 1 + 17.0T + 73T^{2} \)
79 \( 1 - 3.13T + 79T^{2} \)
83 \( 1 - 8.22T + 83T^{2} \)
89 \( 1 + 9.89T + 89T^{2} \)
97 \( 1 - 14.7T + 97T^{2} \)
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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

−7.68714679305717208231123400405, −7.39747818904429612463911215141, −6.13552395417871940419082579343, −5.93364656027160240385681956040, −4.66764473060443797495167573215, −4.21619281443108146878421706784, −3.53279989610734530522788604446, −2.67346039142872270468538284733, −2.18494701443797804958311475323, −0.933933928100366358608002462078, 0.933933928100366358608002462078, 2.18494701443797804958311475323, 2.67346039142872270468538284733, 3.53279989610734530522788604446, 4.21619281443108146878421706784, 4.66764473060443797495167573215, 5.93364656027160240385681956040, 6.13552395417871940419082579343, 7.39747818904429612463911215141, 7.68714679305717208231123400405

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