Properties

Label 2-8470-1.1-c1-0-45
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 − 0.935·3-s + 4-s + 5-s − 0.935·6-s − 7-s + 8-s − 2.12·9-s + 10-s − 0.935·12-s + 2.51·13-s − 14-s − 0.935·15-s + 16-s − 4.68·17-s − 2.12·18-s − 4.05·19-s + 20-s + 0.935·21-s − 3.46·23-s − 0.935·24-s + 25-s + 2.51·26-s + 4.79·27-s − 28-s + 7.46·29-s − 0.935·30-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.540·3-s + 0.5·4-s + 0.447·5-s − 0.381·6-s − 0.377·7-s + 0.353·8-s − 0.708·9-s + 0.316·10-s − 0.270·12-s + 0.697·13-s − 0.267·14-s − 0.241·15-s + 0.250·16-s − 1.13·17-s − 0.500·18-s − 0.930·19-s + 0.223·20-s + 0.204·21-s − 0.723·23-s − 0.190·24-s + 0.200·25-s + 0.493·26-s + 0.922·27-s − 0.188·28-s + 1.38·29-s − 0.170·30-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

\(L(1)\) \(\approx\) \(2.272070568\)
\(L(\frac12)\) \(\approx\) \(2.272070568\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.935T + 3T^{2} \)
13 \( 1 - 2.51T + 13T^{2} \)
17 \( 1 + 4.68T + 17T^{2} \)
19 \( 1 + 4.05T + 19T^{2} \)
23 \( 1 + 3.46T + 23T^{2} \)
29 \( 1 - 7.46T + 29T^{2} \)
31 \( 1 - 1.54T + 31T^{2} \)
37 \( 1 + 4.92T + 37T^{2} \)
41 \( 1 + 1.65T + 41T^{2} \)
43 \( 1 - 7.97T + 43T^{2} \)
47 \( 1 - 1.77T + 47T^{2} \)
53 \( 1 - 7.14T + 53T^{2} \)
59 \( 1 + 0.426T + 59T^{2} \)
61 \( 1 + 6.15T + 61T^{2} \)
67 \( 1 + 5.99T + 67T^{2} \)
71 \( 1 - 6.53T + 71T^{2} \)
73 \( 1 - 0.855T + 73T^{2} \)
79 \( 1 - 9.86T + 79T^{2} \)
83 \( 1 - 8.15T + 83T^{2} \)
89 \( 1 - 1.00T + 89T^{2} \)
97 \( 1 - 11.3T + 97T^{2} \)
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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

−7.66456139123231248643726816662, −6.66130918016688921367633577424, −6.31641331911314994771398990456, −5.85875300401472896389506785568, −5.00449041690932326017125137813, −4.37814344637352028846827025944, −3.55867053297651267761807556094, −2.65862163912308979507733403485, −1.98931834315529945869751247918, −0.65704080240809037703148630982, 0.65704080240809037703148630982, 1.98931834315529945869751247918, 2.65862163912308979507733403485, 3.55867053297651267761807556094, 4.37814344637352028846827025944, 5.00449041690932326017125137813, 5.85875300401472896389506785568, 6.31641331911314994771398990456, 6.66130918016688921367633577424, 7.66456139123231248643726816662

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