Properties

Label 2-8470-1.1-c1-0-68
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 + 1.14·3-s + 4-s + 5-s − 1.14·6-s − 7-s − 8-s − 1.67·9-s − 10-s + 1.14·12-s + 3.79·13-s + 14-s + 1.14·15-s + 16-s + 6.47·17-s + 1.67·18-s + 2.67·19-s + 20-s − 1.14·21-s + 2.65·23-s − 1.14·24-s + 25-s − 3.79·26-s − 5.37·27-s − 28-s + 0.386·29-s − 1.14·30-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.663·3-s + 0.5·4-s + 0.447·5-s − 0.469·6-s − 0.377·7-s − 0.353·8-s − 0.559·9-s − 0.316·10-s + 0.331·12-s + 1.05·13-s + 0.267·14-s + 0.296·15-s + 0.250·16-s + 1.57·17-s + 0.395·18-s + 0.614·19-s + 0.223·20-s − 0.250·21-s + 0.553·23-s − 0.234·24-s + 0.200·25-s − 0.744·26-s − 1.03·27-s − 0.188·28-s + 0.0718·29-s − 0.209·30-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

\(L(1)\) \(\approx\) \(2.115483962\)
\(L(\frac12)\) \(\approx\) \(2.115483962\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.14T + 3T^{2} \)
13 \( 1 - 3.79T + 13T^{2} \)
17 \( 1 - 6.47T + 17T^{2} \)
19 \( 1 - 2.67T + 19T^{2} \)
23 \( 1 - 2.65T + 23T^{2} \)
29 \( 1 - 0.386T + 29T^{2} \)
31 \( 1 + 6.01T + 31T^{2} \)
37 \( 1 - 2.38T + 37T^{2} \)
41 \( 1 - 9.47T + 41T^{2} \)
43 \( 1 + 4.66T + 43T^{2} \)
47 \( 1 + 4.48T + 47T^{2} \)
53 \( 1 + 5.09T + 53T^{2} \)
59 \( 1 - 4.74T + 59T^{2} \)
61 \( 1 - 11.6T + 61T^{2} \)
67 \( 1 - 5.71T + 67T^{2} \)
71 \( 1 - 3.33T + 71T^{2} \)
73 \( 1 - 15.0T + 73T^{2} \)
79 \( 1 + 6.03T + 79T^{2} \)
83 \( 1 + 11.1T + 83T^{2} \)
89 \( 1 + 5.87T + 89T^{2} \)
97 \( 1 + 3.30T + 97T^{2} \)
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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

−8.083696502174804214125877032759, −7.21827148342552633899744190522, −6.53992890884553092718114690588, −5.65543757028184912859262079006, −5.39580343270807929005473190096, −3.91745160870878178667563781698, −3.27862373254277757481007035569, −2.69533593598287245645977617227, −1.66719804482218482819879359091, −0.794641884309409337643998605268, 0.794641884309409337643998605268, 1.66719804482218482819879359091, 2.69533593598287245645977617227, 3.27862373254277757481007035569, 3.91745160870878178667563781698, 5.39580343270807929005473190096, 5.65543757028184912859262079006, 6.53992890884553092718114690588, 7.21827148342552633899744190522, 8.083696502174804214125877032759

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