Properties

Label 2-1470-1.1-c3-0-76
Degree $2$
Conductor $1470$
Sign $-1$
Analytic cond. $86.7328$
Root an. cond. $9.31304$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 3·3-s + 4·4-s − 5·5-s + 6·6-s + 8·8-s + 9·9-s − 10·10-s − 20·11-s + 12·12-s − 26·13-s − 15·15-s + 16·16-s + 26·17-s + 18·18-s + 42·19-s − 20·20-s − 40·22-s − 194·23-s + 24·24-s + 25·25-s − 52·26-s + 27·27-s − 42·29-s − 30·30-s − 274·31-s + 32·32-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s + 0.353·8-s + 1/3·9-s − 0.316·10-s − 0.548·11-s + 0.288·12-s − 0.554·13-s − 0.258·15-s + 1/4·16-s + 0.370·17-s + 0.235·18-s + 0.507·19-s − 0.223·20-s − 0.387·22-s − 1.75·23-s + 0.204·24-s + 1/5·25-s − 0.392·26-s + 0.192·27-s − 0.268·29-s − 0.182·30-s − 1.58·31-s + 0.176·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1470\)    =    \(2 \cdot 3 \cdot 5 \cdot 7^{2}\)
Sign: $-1$
Analytic conductor: \(86.7328\)
Root analytic conductor: \(9.31304\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1470,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - p T \)
3 \( 1 - p T \)
5 \( 1 + p T \)
7 \( 1 \)
good11 \( 1 + 20 T + p^{3} T^{2} \)
13 \( 1 + 2 p T + p^{3} T^{2} \)
17 \( 1 - 26 T + p^{3} T^{2} \)
19 \( 1 - 42 T + p^{3} T^{2} \)
23 \( 1 + 194 T + p^{3} T^{2} \)
29 \( 1 + 42 T + p^{3} T^{2} \)
31 \( 1 + 274 T + p^{3} T^{2} \)
37 \( 1 - 2 T + p^{3} T^{2} \)
41 \( 1 - 250 T + p^{3} T^{2} \)
43 \( 1 + 296 T + p^{3} T^{2} \)
47 \( 1 - 328 T + p^{3} T^{2} \)
53 \( 1 + 148 T + p^{3} T^{2} \)
59 \( 1 + 488 T + p^{3} T^{2} \)
61 \( 1 + 272 T + p^{3} T^{2} \)
67 \( 1 - 8 T + p^{3} T^{2} \)
71 \( 1 + 684 T + p^{3} T^{2} \)
73 \( 1 + 310 T + p^{3} T^{2} \)
79 \( 1 + 584 T + p^{3} T^{2} \)
83 \( 1 + 404 T + p^{3} T^{2} \)
89 \( 1 - 266 T + p^{3} T^{2} \)
97 \( 1 - 678 T + p^{3} T^{2} \)
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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

−8.635936922300886917017529933776, −7.60143256374504873944589335066, −7.44777310923686772253542995322, −6.14495061981749486423518044561, −5.34204030911101157228858345292, −4.37112121332531611957298201928, −3.58928151427466924974371533681, −2.69235687417875711535002437845, −1.67444903260951934795282953886, 0, 1.67444903260951934795282953886, 2.69235687417875711535002437845, 3.58928151427466924974371533681, 4.37112121332531611957298201928, 5.34204030911101157228858345292, 6.14495061981749486423518044561, 7.44777310923686772253542995322, 7.60143256374504873944589335066, 8.635936922300886917017529933776

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