Properties

Label 2-8470-1.1-c1-0-91
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 2.64·3-s + 4-s + 5-s + 2.64·6-s − 7-s − 8-s + 3.99·9-s − 10-s − 2.64·12-s − 3.06·13-s + 14-s − 2.64·15-s + 16-s + 0.0683·17-s − 3.99·18-s − 2.77·19-s + 20-s + 2.64·21-s + 1.83·23-s + 2.64·24-s + 25-s + 3.06·26-s − 2.62·27-s − 28-s − 3.30·29-s + 2.64·30-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.52·3-s + 0.5·4-s + 0.447·5-s + 1.07·6-s − 0.377·7-s − 0.353·8-s + 1.33·9-s − 0.316·10-s − 0.763·12-s − 0.848·13-s + 0.267·14-s − 0.682·15-s + 0.250·16-s + 0.0165·17-s − 0.940·18-s − 0.635·19-s + 0.223·20-s + 0.577·21-s + 0.381·23-s + 0.539·24-s + 0.200·25-s + 0.600·26-s − 0.504·27-s − 0.188·28-s − 0.614·29-s + 0.482·30-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: \(1\)
Selberg data: \((2,\ 8470,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.64T + 3T^{2} \)
13 \( 1 + 3.06T + 13T^{2} \)
17 \( 1 - 0.0683T + 17T^{2} \)
19 \( 1 + 2.77T + 19T^{2} \)
23 \( 1 - 1.83T + 23T^{2} \)
29 \( 1 + 3.30T + 29T^{2} \)
31 \( 1 - 4.12T + 31T^{2} \)
37 \( 1 + 5.63T + 37T^{2} \)
41 \( 1 - 5.13T + 41T^{2} \)
43 \( 1 + 2.84T + 43T^{2} \)
47 \( 1 - 11.9T + 47T^{2} \)
53 \( 1 - 0.123T + 53T^{2} \)
59 \( 1 + 11.1T + 59T^{2} \)
61 \( 1 + 2.53T + 61T^{2} \)
67 \( 1 + 2.20T + 67T^{2} \)
71 \( 1 - 13.7T + 71T^{2} \)
73 \( 1 + 6.98T + 73T^{2} \)
79 \( 1 - 12.4T + 79T^{2} \)
83 \( 1 - 3.05T + 83T^{2} \)
89 \( 1 - 13.5T + 89T^{2} \)
97 \( 1 + 2.74T + 97T^{2} \)
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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

−7.31312261803178279568500265434, −6.62750982913381127067563713541, −6.21828583887259701955731295821, −5.46570443040466944998639085628, −4.92108533357783113758393278848, −4.03237098851567463609847853469, −2.85963721586264389540227475931, −1.97652911839618267154678113838, −0.910397570636313685935187403424, 0, 0.910397570636313685935187403424, 1.97652911839618267154678113838, 2.85963721586264389540227475931, 4.03237098851567463609847853469, 4.92108533357783113758393278848, 5.46570443040466944998639085628, 6.21828583887259701955731295821, 6.62750982913381127067563713541, 7.31312261803178279568500265434

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