Properties

Label 2-8470-1.1-c1-0-162
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 1.73·3-s + 4-s − 5-s − 1.73·6-s − 7-s − 8-s + 10-s + 1.73·12-s − 1.73·13-s + 14-s − 1.73·15-s + 16-s + 4.73·17-s + 1.73·19-s − 20-s − 1.73·21-s − 4.46·23-s − 1.73·24-s + 25-s + 1.73·26-s − 5.19·27-s − 28-s + 9.46·29-s + 1.73·30-s − 6.73·31-s − 32-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.00·3-s + 0.5·4-s − 0.447·5-s − 0.707·6-s − 0.377·7-s − 0.353·8-s + 0.316·10-s + 0.500·12-s − 0.480·13-s + 0.267·14-s − 0.447·15-s + 0.250·16-s + 1.14·17-s + 0.397·19-s − 0.223·20-s − 0.377·21-s − 0.930·23-s − 0.353·24-s + 0.200·25-s + 0.339·26-s − 1.00·27-s − 0.188·28-s + 1.75·29-s + 0.316·30-s − 1.20·31-s − 0.176·32-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: \(1\)
Selberg data: \((2,\ 8470,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.73T + 3T^{2} \)
13 \( 1 + 1.73T + 13T^{2} \)
17 \( 1 - 4.73T + 17T^{2} \)
19 \( 1 - 1.73T + 19T^{2} \)
23 \( 1 + 4.46T + 23T^{2} \)
29 \( 1 - 9.46T + 29T^{2} \)
31 \( 1 + 6.73T + 31T^{2} \)
37 \( 1 + 0.535T + 37T^{2} \)
41 \( 1 + 3.26T + 41T^{2} \)
43 \( 1 - 9.66T + 43T^{2} \)
47 \( 1 - 0.732T + 47T^{2} \)
53 \( 1 + 4.73T + 53T^{2} \)
59 \( 1 + T + 59T^{2} \)
61 \( 1 - 4T + 61T^{2} \)
67 \( 1 - 3.80T + 67T^{2} \)
71 \( 1 + 10.9T + 71T^{2} \)
73 \( 1 + 7.66T + 73T^{2} \)
79 \( 1 + 13.9T + 79T^{2} \)
83 \( 1 + 1.92T + 83T^{2} \)
89 \( 1 + 1.26T + 89T^{2} \)
97 \( 1 + 5.46T + 97T^{2} \)
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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

−7.50739695353278018964179195550, −7.19388363328662825066792092611, −6.13092994185035302680127168585, −5.53208802432604453052306188894, −4.46089811064048702901916626141, −3.57134596045895562263583275975, −3.01010737437219644387984894911, −2.31259869164806212995133579681, −1.23608780910489650602327876673, 0, 1.23608780910489650602327876673, 2.31259869164806212995133579681, 3.01010737437219644387984894911, 3.57134596045895562263583275975, 4.46089811064048702901916626141, 5.53208802432604453052306188894, 6.13092994185035302680127168585, 7.19388363328662825066792092611, 7.50739695353278018964179195550

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