Properties

Label 2-1380-1.1-c1-0-14
Degree $2$
Conductor $1380$
Sign $-1$
Analytic cond. $11.0193$
Root an. cond. $3.31954$
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 − 7-s + 9-s − 4·13-s − 15-s − 3·17-s − 4·19-s − 21-s − 23-s + 25-s + 27-s − 3·29-s − 7·31-s + 35-s + 11·37-s − 4·39-s − 9·41-s − 4·43-s − 45-s + 6·47-s − 6·49-s − 3·51-s + 9·53-s − 4·57-s + 3·59-s − 10·61-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.447·5-s − 0.377·7-s + 1/3·9-s − 1.10·13-s − 0.258·15-s − 0.727·17-s − 0.917·19-s − 0.218·21-s − 0.208·23-s + 1/5·25-s + 0.192·27-s − 0.557·29-s − 1.25·31-s + 0.169·35-s + 1.80·37-s − 0.640·39-s − 1.40·41-s − 0.609·43-s − 0.149·45-s + 0.875·47-s − 6/7·49-s − 0.420·51-s + 1.23·53-s − 0.529·57-s + 0.390·59-s − 1.28·61-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: yes
Arithmetic: yes
Character: $\chi_{1380} (1, \cdot )$
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 \( 1 + T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 3 T + p T^{2} \)
31 \( 1 + 7 T + p T^{2} \)
37 \( 1 - 11 T + p T^{2} \)
41 \( 1 + 9 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 - 9 T + p T^{2} \)
59 \( 1 - 3 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 13 T + p T^{2} \)
71 \( 1 - 9 T + p T^{2} \)
73 \( 1 + 16 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 15 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−9.136723042105678626497401946106, −8.426712326417971032599906101216, −7.52652311249344567844634763071, −6.93543506214015159115646955058, −5.92285958674440455094839040519, −4.74028235498131449962795690414, −3.99337842949862406268567434883, −2.93143144537160068866262117724, −1.95878492833586522695243535909, 0, 1.95878492833586522695243535909, 2.93143144537160068866262117724, 3.99337842949862406268567434883, 4.74028235498131449962795690414, 5.92285958674440455094839040519, 6.93543506214015159115646955058, 7.52652311249344567844634763071, 8.426712326417971032599906101216, 9.136723042105678626497401946106

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