Properties

Label 2-1320-1.1-c1-0-18
Degree $2$
Conductor $1320$
Sign $-1$
Analytic cond. $10.5402$
Root an. cond. $3.24657$
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  + 3-s − 5-s − 2·7-s + 9-s + 11-s − 15-s − 8·17-s − 8·19-s − 2·21-s + 4·23-s + 25-s + 27-s − 6·29-s + 33-s + 2·35-s + 6·37-s − 2·41-s + 2·43-s − 45-s − 4·47-s − 3·49-s − 8·51-s − 2·53-s − 55-s − 8·57-s − 12·59-s − 6·61-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.447·5-s − 0.755·7-s + 1/3·9-s + 0.301·11-s − 0.258·15-s − 1.94·17-s − 1.83·19-s − 0.436·21-s + 0.834·23-s + 1/5·25-s + 0.192·27-s − 1.11·29-s + 0.174·33-s + 0.338·35-s + 0.986·37-s − 0.312·41-s + 0.304·43-s − 0.149·45-s − 0.583·47-s − 3/7·49-s − 1.12·51-s − 0.274·53-s − 0.134·55-s − 1.05·57-s − 1.56·59-s − 0.768·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1320\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 11\)
Sign: $-1$
Analytic conductor: \(10.5402\)
Root analytic conductor: \(3.24657\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1320} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1320,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
11 \( 1 - T \)
good7 \( 1 + 2 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 + 8 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 8 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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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

−9.027334298425479681893096209206, −8.651401501193602054770305574930, −7.61356529915864130916723689853, −6.71839284015280032981724873296, −6.20212391078145141501314845131, −4.66727813556072448891583848456, −4.04072250068062845131870077954, −2.98669163959032499870715553371, −1.95760264932402532312970050724, 0, 1.95760264932402532312970050724, 2.98669163959032499870715553371, 4.04072250068062845131870077954, 4.66727813556072448891583848456, 6.20212391078145141501314845131, 6.71839284015280032981724873296, 7.61356529915864130916723689853, 8.651401501193602054770305574930, 9.027334298425479681893096209206

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