Properties

Label 2-3360-1.1-c1-0-33
Degree $2$
Conductor $3360$
Sign $-1$
Analytic cond. $26.8297$
Root an. cond. $5.17974$
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  − 3-s − 5-s + 7-s + 9-s + 1.46·11-s − 3.46·13-s + 15-s + 2·17-s + 1.46·19-s − 21-s − 6.92·23-s + 25-s − 27-s − 2·29-s − 1.46·31-s − 1.46·33-s − 35-s + 2·37-s + 3.46·39-s + 8.92·41-s − 4·43-s − 45-s + 2.92·47-s + 49-s − 2·51-s − 7.46·53-s − 1.46·55-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.447·5-s + 0.377·7-s + 0.333·9-s + 0.441·11-s − 0.960·13-s + 0.258·15-s + 0.485·17-s + 0.335·19-s − 0.218·21-s − 1.44·23-s + 0.200·25-s − 0.192·27-s − 0.371·29-s − 0.262·31-s − 0.254·33-s − 0.169·35-s + 0.328·37-s + 0.554·39-s + 1.39·41-s − 0.609·43-s − 0.149·45-s + 0.427·47-s + 0.142·49-s − 0.280·51-s − 1.02·53-s − 0.197·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3360\)    =    \(2^{5} \cdot 3 \cdot 5 \cdot 7\)
Sign: $-1$
Analytic conductor: \(26.8297\)
Root analytic conductor: \(5.17974\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3360} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 3360,\ (\ :1/2),\ -1)\)

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 \( 1 \)
3 \( 1 + T \)
5 \( 1 + T \)
7 \( 1 - T \)
good11 \( 1 - 1.46T + 11T^{2} \)
13 \( 1 + 3.46T + 13T^{2} \)
17 \( 1 - 2T + 17T^{2} \)
19 \( 1 - 1.46T + 19T^{2} \)
23 \( 1 + 6.92T + 23T^{2} \)
29 \( 1 + 2T + 29T^{2} \)
31 \( 1 + 1.46T + 31T^{2} \)
37 \( 1 - 2T + 37T^{2} \)
41 \( 1 - 8.92T + 41T^{2} \)
43 \( 1 + 4T + 43T^{2} \)
47 \( 1 - 2.92T + 47T^{2} \)
53 \( 1 + 7.46T + 53T^{2} \)
59 \( 1 + 13.8T + 59T^{2} \)
61 \( 1 - 8.92T + 61T^{2} \)
67 \( 1 + 6.92T + 67T^{2} \)
71 \( 1 - 4.39T + 71T^{2} \)
73 \( 1 - 7.46T + 73T^{2} \)
79 \( 1 + 4T + 79T^{2} \)
83 \( 1 + 1.07T + 83T^{2} \)
89 \( 1 - 0.928T + 89T^{2} \)
97 \( 1 - 12.5T + 97T^{2} \)
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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

−7.918892867974253287508352781512, −7.68293300708513972525411426616, −6.74509003335604703790601718423, −5.95329657718951798907588315836, −5.19284306339427619044909183085, −4.40567215193202450495172573143, −3.68021303297323414908734447689, −2.48728453139799749290264788725, −1.35264141809480354806388811188, 0, 1.35264141809480354806388811188, 2.48728453139799749290264788725, 3.68021303297323414908734447689, 4.40567215193202450495172573143, 5.19284306339427619044909183085, 5.95329657718951798907588315836, 6.74509003335604703790601718423, 7.68293300708513972525411426616, 7.918892867974253287508352781512

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