Properties

Label 2-8470-1.1-c1-0-122
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 + 2.18·3-s + 4-s + 5-s + 2.18·6-s − 7-s + 8-s + 1.76·9-s + 10-s + 2.18·12-s − 2.53·13-s − 14-s + 2.18·15-s + 16-s + 5.73·17-s + 1.76·18-s + 2.24·19-s + 20-s − 2.18·21-s − 2.51·23-s + 2.18·24-s + 25-s − 2.53·26-s − 2.69·27-s − 28-s + 8.61·29-s + 2.18·30-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.26·3-s + 0.5·4-s + 0.447·5-s + 0.891·6-s − 0.377·7-s + 0.353·8-s + 0.588·9-s + 0.316·10-s + 0.630·12-s − 0.702·13-s − 0.267·14-s + 0.563·15-s + 0.250·16-s + 1.39·17-s + 0.416·18-s + 0.514·19-s + 0.223·20-s − 0.476·21-s − 0.524·23-s + 0.445·24-s + 0.200·25-s − 0.496·26-s − 0.518·27-s − 0.188·28-s + 1.60·29-s + 0.398·30-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

\(L(1)\) \(\approx\) \(5.878583527\)
\(L(\frac12)\) \(\approx\) \(5.878583527\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.18T + 3T^{2} \)
13 \( 1 + 2.53T + 13T^{2} \)
17 \( 1 - 5.73T + 17T^{2} \)
19 \( 1 - 2.24T + 19T^{2} \)
23 \( 1 + 2.51T + 23T^{2} \)
29 \( 1 - 8.61T + 29T^{2} \)
31 \( 1 - 1.84T + 31T^{2} \)
37 \( 1 + 4.63T + 37T^{2} \)
41 \( 1 - 10.5T + 41T^{2} \)
43 \( 1 - 2.18T + 43T^{2} \)
47 \( 1 - 3.22T + 47T^{2} \)
53 \( 1 + 4.01T + 53T^{2} \)
59 \( 1 - 2.01T + 59T^{2} \)
61 \( 1 - 0.268T + 61T^{2} \)
67 \( 1 - 3.86T + 67T^{2} \)
71 \( 1 + 9.55T + 71T^{2} \)
73 \( 1 - 11.4T + 73T^{2} \)
79 \( 1 - 11.7T + 79T^{2} \)
83 \( 1 - 7.34T + 83T^{2} \)
89 \( 1 + 2.65T + 89T^{2} \)
97 \( 1 - 16.5T + 97T^{2} \)
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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

−7.80370986418016181202997338119, −7.18163080305679110838276200282, −6.36282338987220644609203917116, −5.66687815858713794206763972541, −4.98012570500814712533038360704, −4.09207224205577870995241042793, −3.34337134544425258835176913886, −2.77541316228050721768010033430, −2.16967715942033118624295863563, −1.03338901485360479204673780015, 1.03338901485360479204673780015, 2.16967715942033118624295863563, 2.77541316228050721768010033430, 3.34337134544425258835176913886, 4.09207224205577870995241042793, 4.98012570500814712533038360704, 5.66687815858713794206763972541, 6.36282338987220644609203917116, 7.18163080305679110838276200282, 7.80370986418016181202997338119

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