Properties

Label 2-8470-1.1-c1-0-75
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 + 1.22·3-s + 4-s − 5-s + 1.22·6-s − 7-s + 8-s − 1.50·9-s − 10-s + 1.22·12-s + 3.47·13-s − 14-s − 1.22·15-s + 16-s − 0.138·17-s − 1.50·18-s + 0.515·19-s − 20-s − 1.22·21-s + 1.51·23-s + 1.22·24-s + 25-s + 3.47·26-s − 5.50·27-s − 28-s + 9.24·29-s − 1.22·30-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.705·3-s + 0.5·4-s − 0.447·5-s + 0.498·6-s − 0.377·7-s + 0.353·8-s − 0.502·9-s − 0.316·10-s + 0.352·12-s + 0.964·13-s − 0.267·14-s − 0.315·15-s + 0.250·16-s − 0.0335·17-s − 0.355·18-s + 0.118·19-s − 0.223·20-s − 0.266·21-s + 0.314·23-s + 0.249·24-s + 0.200·25-s + 0.682·26-s − 1.05·27-s − 0.188·28-s + 1.71·29-s − 0.223·30-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

\(L(1)\) \(\approx\) \(3.778753322\)
\(L(\frac12)\) \(\approx\) \(3.778753322\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.22T + 3T^{2} \)
13 \( 1 - 3.47T + 13T^{2} \)
17 \( 1 + 0.138T + 17T^{2} \)
19 \( 1 - 0.515T + 19T^{2} \)
23 \( 1 - 1.51T + 23T^{2} \)
29 \( 1 - 9.24T + 29T^{2} \)
31 \( 1 - 10.2T + 31T^{2} \)
37 \( 1 - 1.66T + 37T^{2} \)
41 \( 1 + 12.1T + 41T^{2} \)
43 \( 1 + 10.6T + 43T^{2} \)
47 \( 1 + 1.80T + 47T^{2} \)
53 \( 1 - 1.39T + 53T^{2} \)
59 \( 1 - 9.30T + 59T^{2} \)
61 \( 1 - 13.5T + 61T^{2} \)
67 \( 1 + 1.84T + 67T^{2} \)
71 \( 1 + 6.39T + 71T^{2} \)
73 \( 1 + 11.0T + 73T^{2} \)
79 \( 1 - 9.98T + 79T^{2} \)
83 \( 1 - 2.20T + 83T^{2} \)
89 \( 1 - 1.74T + 89T^{2} \)
97 \( 1 + 4.41T + 97T^{2} \)
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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

−7.949178428680173809592899041793, −6.84216981388300107888496945595, −6.54694871432075301679610392404, −5.69159697899526825236299030646, −4.89166169160210478180116501339, −4.18156850576733662566139553097, −3.23646563288345857764497732961, −3.08316058595281667051443601452, −2.01951979470211914309428645931, −0.823221426125558794979795274433, 0.823221426125558794979795274433, 2.01951979470211914309428645931, 3.08316058595281667051443601452, 3.23646563288345857764497732961, 4.18156850576733662566139553097, 4.89166169160210478180116501339, 5.69159697899526825236299030646, 6.54694871432075301679610392404, 6.84216981388300107888496945595, 7.949178428680173809592899041793

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