Properties

Label 2-8470-1.1-c1-0-47
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 + 0.0970·3-s + 4-s + 5-s − 0.0970·6-s + 7-s − 8-s − 2.99·9-s − 10-s + 0.0970·12-s + 4.32·13-s − 14-s + 0.0970·15-s + 16-s − 2.12·17-s + 2.99·18-s − 5.51·19-s + 20-s + 0.0970·21-s + 0.922·23-s − 0.0970·24-s + 25-s − 4.32·26-s − 0.581·27-s + 28-s + 8.02·29-s − 0.0970·30-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.0560·3-s + 0.5·4-s + 0.447·5-s − 0.0396·6-s + 0.377·7-s − 0.353·8-s − 0.996·9-s − 0.316·10-s + 0.0280·12-s + 1.19·13-s − 0.267·14-s + 0.0250·15-s + 0.250·16-s − 0.514·17-s + 0.704·18-s − 1.26·19-s + 0.223·20-s + 0.0211·21-s + 0.192·23-s − 0.0198·24-s + 0.200·25-s − 0.848·26-s − 0.111·27-s + 0.188·28-s + 1.49·29-s − 0.0177·30-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

\(L(1)\) \(\approx\) \(1.474309693\)
\(L(\frac12)\) \(\approx\) \(1.474309693\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.0970T + 3T^{2} \)
13 \( 1 - 4.32T + 13T^{2} \)
17 \( 1 + 2.12T + 17T^{2} \)
19 \( 1 + 5.51T + 19T^{2} \)
23 \( 1 - 0.922T + 23T^{2} \)
29 \( 1 - 8.02T + 29T^{2} \)
31 \( 1 - 1.21T + 31T^{2} \)
37 \( 1 + 5.01T + 37T^{2} \)
41 \( 1 - 1.92T + 41T^{2} \)
43 \( 1 + 3.75T + 43T^{2} \)
47 \( 1 + 0.865T + 47T^{2} \)
53 \( 1 + 0.729T + 53T^{2} \)
59 \( 1 + 0.178T + 59T^{2} \)
61 \( 1 - 13.0T + 61T^{2} \)
67 \( 1 + 3.55T + 67T^{2} \)
71 \( 1 - 5.96T + 71T^{2} \)
73 \( 1 - 14.8T + 73T^{2} \)
79 \( 1 + 13.1T + 79T^{2} \)
83 \( 1 + 2.63T + 83T^{2} \)
89 \( 1 + 18.4T + 89T^{2} \)
97 \( 1 - 0.0855T + 97T^{2} \)
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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

−8.160075381376608599470349136039, −7.01649275359835722235543013150, −6.45516348183814286817245293121, −5.91876211995740533488658638841, −5.12789122208833622471543813452, −4.24259366254694932448373950986, −3.29509546433392112248891116559, −2.47959415152666095479275985882, −1.72692576764083500861062311709, −0.65964727515123509710315293016, 0.65964727515123509710315293016, 1.72692576764083500861062311709, 2.47959415152666095479275985882, 3.29509546433392112248891116559, 4.24259366254694932448373950986, 5.12789122208833622471543813452, 5.91876211995740533488658638841, 6.45516348183814286817245293121, 7.01649275359835722235543013150, 8.160075381376608599470349136039

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