Properties

Label 2-1380-1.1-c1-0-12
Degree $2$
Conductor $1380$
Sign $-1$
Analytic cond. $11.0193$
Root an. cond. $3.31954$
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 − 2.46·7-s + 9-s + 4.19·11-s − 3.26·13-s − 15-s − 7.73·17-s + 0.732·19-s + 2.46·21-s − 23-s + 25-s − 27-s + 7.19·29-s − 31-s − 4.19·33-s − 2.46·35-s − 11.3·37-s + 3.26·39-s − 7.73·41-s + 3.46·43-s + 45-s + 0.732·47-s − 0.928·49-s + 7.73·51-s + 6.66·53-s + 4.19·55-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.447·5-s − 0.931·7-s + 0.333·9-s + 1.26·11-s − 0.906·13-s − 0.258·15-s − 1.87·17-s + 0.167·19-s + 0.537·21-s − 0.208·23-s + 0.200·25-s − 0.192·27-s + 1.33·29-s − 0.179·31-s − 0.730·33-s − 0.416·35-s − 1.87·37-s + 0.523·39-s − 1.20·41-s + 0.528·43-s + 0.149·45-s + 0.106·47-s − 0.132·49-s + 1.08·51-s + 0.914·53-s + 0.565·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1380\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 23\)
Sign: $-1$
Analytic conductor: \(11.0193\)
Root analytic conductor: \(3.31954\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1380,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
23 \( 1 + T \)
good7 \( 1 + 2.46T + 7T^{2} \)
11 \( 1 - 4.19T + 11T^{2} \)
13 \( 1 + 3.26T + 13T^{2} \)
17 \( 1 + 7.73T + 17T^{2} \)
19 \( 1 - 0.732T + 19T^{2} \)
29 \( 1 - 7.19T + 29T^{2} \)
31 \( 1 + T + 31T^{2} \)
37 \( 1 + 11.3T + 37T^{2} \)
41 \( 1 + 7.73T + 41T^{2} \)
43 \( 1 - 3.46T + 43T^{2} \)
47 \( 1 - 0.732T + 47T^{2} \)
53 \( 1 - 6.66T + 53T^{2} \)
59 \( 1 + 7.19T + 59T^{2} \)
61 \( 1 + 10.7T + 61T^{2} \)
67 \( 1 - 5T + 67T^{2} \)
71 \( 1 + 14.1T + 71T^{2} \)
73 \( 1 + 11.2T + 73T^{2} \)
79 \( 1 - 4T + 79T^{2} \)
83 \( 1 + 6.66T + 83T^{2} \)
89 \( 1 + 16.3T + 89T^{2} \)
97 \( 1 - 4T + 97T^{2} \)
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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

−9.182603711375795814197625889171, −8.678853847709236662138217628469, −7.15867901871030863218743231845, −6.67943667020700224682384780074, −6.05835205408259222625321645976, −4.93306058587191971028364748992, −4.13532139823804113200282477183, −2.92950330326600493566813392170, −1.69213710279545812099123622359, 0, 1.69213710279545812099123622359, 2.92950330326600493566813392170, 4.13532139823804113200282477183, 4.93306058587191971028364748992, 6.05835205408259222625321645976, 6.67943667020700224682384780074, 7.15867901871030863218743231845, 8.678853847709236662138217628469, 9.182603711375795814197625889171

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