Properties

Label 2-1380-1.1-c1-0-3
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 5-s − 3.73·7-s + 9-s + 4.84·11-s − 2.84·13-s − 15-s + 0.890·17-s + 6.84·19-s − 3.73·21-s + 23-s + 25-s + 27-s − 0.890·29-s + 7.73·31-s + 4.84·33-s + 3.73·35-s − 1.95·37-s − 2.84·39-s + 12.3·41-s − 3.47·43-s − 45-s − 6.62·47-s + 6.95·49-s + 0.890·51-s + 12.3·53-s − 4.84·55-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.447·5-s − 1.41·7-s + 0.333·9-s + 1.46·11-s − 0.788·13-s − 0.258·15-s + 0.216·17-s + 1.57·19-s − 0.815·21-s + 0.208·23-s + 0.200·25-s + 0.192·27-s − 0.165·29-s + 1.38·31-s + 0.843·33-s + 0.631·35-s − 0.321·37-s − 0.455·39-s + 1.93·41-s − 0.529·43-s − 0.149·45-s − 0.966·47-s + 0.993·49-s + 0.124·51-s + 1.69·53-s − 0.653·55-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: $\chi_{1380} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1380,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.750599456\)
\(L(\frac12)\) \(\approx\) \(1.750599456\)
\(L(\frac{3}{2})\) 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 \( 1 + 3.73T + 7T^{2} \)
11 \( 1 - 4.84T + 11T^{2} \)
13 \( 1 + 2.84T + 13T^{2} \)
17 \( 1 - 0.890T + 17T^{2} \)
19 \( 1 - 6.84T + 19T^{2} \)
29 \( 1 + 0.890T + 29T^{2} \)
31 \( 1 - 7.73T + 31T^{2} \)
37 \( 1 + 1.95T + 37T^{2} \)
41 \( 1 - 12.3T + 41T^{2} \)
43 \( 1 + 3.47T + 43T^{2} \)
47 \( 1 + 6.62T + 47T^{2} \)
53 \( 1 - 12.3T + 53T^{2} \)
59 \( 1 - 0.890T + 59T^{2} \)
61 \( 1 - 8.62T + 61T^{2} \)
67 \( 1 - 7.73T + 67T^{2} \)
71 \( 1 + 12.3T + 71T^{2} \)
73 \( 1 - 16.5T + 73T^{2} \)
79 \( 1 - 13.4T + 79T^{2} \)
83 \( 1 + 4.58T + 83T^{2} \)
89 \( 1 + 15.1T + 89T^{2} \)
97 \( 1 + 17.2T + 97T^{2} \)
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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

−9.689468344054512375997941584647, −8.932435827041407141097634525861, −7.953559415058402442644368898593, −7.04304329945705091762746347712, −6.57819038218311113684043615644, −5.43287411975762953409106091050, −4.19296708160366480358094398543, −3.45889390362862916763654605637, −2.65190881754065178858108018149, −0.960568129185171567134975236221, 0.960568129185171567134975236221, 2.65190881754065178858108018149, 3.45889390362862916763654605637, 4.19296708160366480358094398543, 5.43287411975762953409106091050, 6.57819038218311113684043615644, 7.04304329945705091762746347712, 7.953559415058402442644368898593, 8.932435827041407141097634525861, 9.689468344054512375997941584647

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