Properties

Label 2-8470-1.1-c1-0-65
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 + 0.245·3-s + 4-s + 5-s + 0.245·6-s − 7-s + 8-s − 2.93·9-s + 10-s + 0.245·12-s + 1.15·13-s − 14-s + 0.245·15-s + 16-s − 6.55·17-s − 2.93·18-s + 3.43·19-s + 20-s − 0.245·21-s + 8.30·23-s + 0.245·24-s + 25-s + 1.15·26-s − 1.46·27-s − 28-s + 2.93·29-s + 0.245·30-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.141·3-s + 0.5·4-s + 0.447·5-s + 0.100·6-s − 0.377·7-s + 0.353·8-s − 0.979·9-s + 0.316·10-s + 0.0709·12-s + 0.319·13-s − 0.267·14-s + 0.0634·15-s + 0.250·16-s − 1.58·17-s − 0.692·18-s + 0.787·19-s + 0.223·20-s − 0.0536·21-s + 1.73·23-s + 0.0501·24-s + 0.200·25-s + 0.225·26-s − 0.281·27-s − 0.188·28-s + 0.544·29-s + 0.0448·30-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

\(L(1)\) \(\approx\) \(3.322177395\)
\(L(\frac12)\) \(\approx\) \(3.322177395\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.245T + 3T^{2} \)
13 \( 1 - 1.15T + 13T^{2} \)
17 \( 1 + 6.55T + 17T^{2} \)
19 \( 1 - 3.43T + 19T^{2} \)
23 \( 1 - 8.30T + 23T^{2} \)
29 \( 1 - 2.93T + 29T^{2} \)
31 \( 1 + 6.33T + 31T^{2} \)
37 \( 1 + 3.86T + 37T^{2} \)
41 \( 1 - 10.5T + 41T^{2} \)
43 \( 1 - 2.95T + 43T^{2} \)
47 \( 1 + 6.17T + 47T^{2} \)
53 \( 1 - 5.35T + 53T^{2} \)
59 \( 1 + 2.24T + 59T^{2} \)
61 \( 1 - 9.98T + 61T^{2} \)
67 \( 1 - 14.2T + 67T^{2} \)
71 \( 1 + 0.380T + 71T^{2} \)
73 \( 1 - 7.23T + 73T^{2} \)
79 \( 1 - 2.27T + 79T^{2} \)
83 \( 1 - 11.3T + 83T^{2} \)
89 \( 1 + 4.48T + 89T^{2} \)
97 \( 1 - 3.45T + 97T^{2} \)
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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

−7.67174401280577422227698608154, −6.78650006491901799268852809179, −6.51286892859433851134860286731, −5.50465537795742171276375792290, −5.21553540867865141059257930141, −4.23942908112130453324766963224, −3.41886188988989925864085183041, −2.74231893219376353227023579841, −2.07222412732177558112603069991, −0.77828356348591121645581550133, 0.77828356348591121645581550133, 2.07222412732177558112603069991, 2.74231893219376353227023579841, 3.41886188988989925864085183041, 4.23942908112130453324766963224, 5.21553540867865141059257930141, 5.50465537795742171276375792290, 6.51286892859433851134860286731, 6.78650006491901799268852809179, 7.67174401280577422227698608154

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