Properties

Label 2-8470-1.1-c1-0-212
Degree $2$
Conductor $8470$
Sign $-1$
Analytic cond. $67.6332$
Root an. cond. $8.22394$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 3-s + 4-s + 5-s + 6-s + 7-s + 8-s − 2·9-s + 10-s + 12-s − 3·13-s + 14-s + 15-s + 16-s − 2·18-s − 3·19-s + 20-s + 21-s − 7·23-s + 24-s + 25-s − 3·26-s − 5·27-s + 28-s − 8·29-s + 30-s + 4·31-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s + 0.377·7-s + 0.353·8-s − 2/3·9-s + 0.316·10-s + 0.288·12-s − 0.832·13-s + 0.267·14-s + 0.258·15-s + 1/4·16-s − 0.471·18-s − 0.688·19-s + 0.223·20-s + 0.218·21-s − 1.45·23-s + 0.204·24-s + 1/5·25-s − 0.588·26-s − 0.962·27-s + 0.188·28-s − 1.48·29-s + 0.182·30-s + 0.718·31-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: yes
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 \( 1 - T + p T^{2} \)
13 \( 1 + 3 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + 3 T + p T^{2} \)
23 \( 1 + 7 T + p T^{2} \)
29 \( 1 + 8 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 + 6 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 + T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 8 T + p T^{2} \)
79 \( 1 - 7 T + p T^{2} \)
83 \( 1 + 9 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 8 T + p T^{2} \)
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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

−7.56956349878490709932890351525, −6.58370590998163550784883792930, −6.02042674178527221767538285076, −5.32142514339069083782747912039, −4.65834116777093687156047561025, −3.83869775733911682966905909240, −3.09175754413920484474899808539, −2.23889039164399094060545668960, −1.76456094872744607430627107044, 0, 1.76456094872744607430627107044, 2.23889039164399094060545668960, 3.09175754413920484474899808539, 3.83869775733911682966905909240, 4.65834116777093687156047561025, 5.32142514339069083782747912039, 6.02042674178527221767538285076, 6.58370590998163550784883792930, 7.56956349878490709932890351525

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