Properties

Label 2-13860-1.1-c1-0-19
Degree $2$
Conductor $13860$
Sign $-1$
Analytic cond. $110.672$
Root an. cond. $10.5201$
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  − 5-s + 7-s + 11-s + 4·13-s − 8·17-s − 4·19-s − 2·23-s + 25-s + 6·29-s − 4·31-s − 35-s + 10·37-s − 2·41-s + 4·43-s − 2·47-s + 49-s − 2·53-s − 55-s − 2·61-s − 4·65-s + 14·67-s + 4·71-s − 16·73-s + 77-s − 8·79-s + 4·83-s + 8·85-s + ⋯
L(s)  = 1  − 0.447·5-s + 0.377·7-s + 0.301·11-s + 1.10·13-s − 1.94·17-s − 0.917·19-s − 0.417·23-s + 1/5·25-s + 1.11·29-s − 0.718·31-s − 0.169·35-s + 1.64·37-s − 0.312·41-s + 0.609·43-s − 0.291·47-s + 1/7·49-s − 0.274·53-s − 0.134·55-s − 0.256·61-s − 0.496·65-s + 1.71·67-s + 0.474·71-s − 1.87·73-s + 0.113·77-s − 0.900·79-s + 0.439·83-s + 0.867·85-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 13860 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13860 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(13860\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 11\)
Sign: $-1$
Analytic conductor: \(110.672\)
Root analytic conductor: \(10.5201\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{13860} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 13860,\ (\ :1/2),\ -1)\)

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 \( 1 \)
5 \( 1 + T \)
7 \( 1 - T \)
11 \( 1 - T \)
good13 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 8 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 2 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 2 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 14 T + p T^{2} \)
71 \( 1 - 4 T + p T^{2} \)
73 \( 1 + 16 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 2 T + p T^{2} \)
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

−16.22572805143233, −15.97117981382975, −15.30329420466120, −14.88538259562536, −14.20230527837253, −13.64392645798186, −13.02212711047218, −12.64949937599626, −11.76187058488310, −11.27734313258495, −10.91649882144861, −10.32514305994610, −9.413066413358029, −8.837508632158390, −8.385162412157134, −7.855667022653770, −6.934469144278041, −6.455049626977213, −5.919801104832243, −4.919217779144033, −4.221336789804846, −3.940646649730927, −2.819016038230910, −2.084112754549061, −1.153007969504626, 0, 1.153007969504626, 2.084112754549061, 2.819016038230910, 3.940646649730927, 4.221336789804846, 4.919217779144033, 5.919801104832243, 6.455049626977213, 6.934469144278041, 7.855667022653770, 8.385162412157134, 8.837508632158390, 9.413066413358029, 10.32514305994610, 10.91649882144861, 11.27734313258495, 11.76187058488310, 12.64949937599626, 13.02212711047218, 13.64392645798186, 14.20230527837253, 14.88538259562536, 15.30329420466120, 15.97117981382975, 16.22572805143233

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