Properties

Label 2-8470-1.1-c1-0-133
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 + 1.61·3-s + 4-s + 5-s + 1.61·6-s + 7-s + 8-s − 0.381·9-s + 10-s + 1.61·12-s + 5.61·13-s + 14-s + 1.61·15-s + 16-s + 4.61·17-s − 0.381·18-s − 5.23·19-s + 20-s + 1.61·21-s + 8.47·23-s + 1.61·24-s + 25-s + 5.61·26-s − 5.47·27-s + 28-s − 5.85·29-s + 1.61·30-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.934·3-s + 0.5·4-s + 0.447·5-s + 0.660·6-s + 0.377·7-s + 0.353·8-s − 0.127·9-s + 0.316·10-s + 0.467·12-s + 1.55·13-s + 0.267·14-s + 0.417·15-s + 0.250·16-s + 1.12·17-s − 0.0900·18-s − 1.20·19-s + 0.223·20-s + 0.353·21-s + 1.76·23-s + 0.330·24-s + 0.200·25-s + 1.10·26-s − 1.05·27-s + 0.188·28-s − 1.08·29-s + 0.295·30-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

\(L(1)\) \(\approx\) \(6.038972720\)
\(L(\frac12)\) \(\approx\) \(6.038972720\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.61T + 3T^{2} \)
13 \( 1 - 5.61T + 13T^{2} \)
17 \( 1 - 4.61T + 17T^{2} \)
19 \( 1 + 5.23T + 19T^{2} \)
23 \( 1 - 8.47T + 23T^{2} \)
29 \( 1 + 5.85T + 29T^{2} \)
31 \( 1 - 5.23T + 31T^{2} \)
37 \( 1 + 3.70T + 37T^{2} \)
41 \( 1 - 8.94T + 41T^{2} \)
43 \( 1 - 3.23T + 43T^{2} \)
47 \( 1 + 13.0T + 47T^{2} \)
53 \( 1 - 6.47T + 53T^{2} \)
59 \( 1 - 4.76T + 59T^{2} \)
61 \( 1 + 11.7T + 61T^{2} \)
67 \( 1 + 1.23T + 67T^{2} \)
71 \( 1 + 7.32T + 71T^{2} \)
73 \( 1 - 4.32T + 73T^{2} \)
79 \( 1 + 0.854T + 79T^{2} \)
83 \( 1 - 13.6T + 83T^{2} \)
89 \( 1 - 0.291T + 89T^{2} \)
97 \( 1 + 3.38T + 97T^{2} \)
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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

−7.87870809529065179562770452508, −7.08178882768162068654961551740, −6.24561125044843219825682568573, −5.75273124229910110061907897565, −4.99561658363045354947728114628, −4.11138758040649833828103682017, −3.41167426098761489453268976011, −2.84490085758486326607227738628, −1.93819093396794543682138531185, −1.11766240293206494176738132138, 1.11766240293206494176738132138, 1.93819093396794543682138531185, 2.84490085758486326607227738628, 3.41167426098761489453268976011, 4.11138758040649833828103682017, 4.99561658363045354947728114628, 5.75273124229910110061907897565, 6.24561125044843219825682568573, 7.08178882768162068654961551740, 7.87870809529065179562770452508

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