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Properties

Label 2-13860-1.1-c1-0-18
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 − 3·17-s − 7·19-s + 9·23-s + 25-s + 3·29-s + 2·31-s − 35-s − 4·37-s + 6·41-s − 43-s + 6·47-s + 49-s − 3·53-s − 55-s + 9·59-s − 61-s + 4·65-s − 10·67-s − 6·71-s + 8·73-s + 77-s − 4·79-s − 3·83-s + ⋯
L(s)  = 1  − 0.447·5-s + 0.377·7-s + 0.301·11-s − 1.10·13-s − 0.727·17-s − 1.60·19-s + 1.87·23-s + 1/5·25-s + 0.557·29-s + 0.359·31-s − 0.169·35-s − 0.657·37-s + 0.937·41-s − 0.152·43-s + 0.875·47-s + 1/7·49-s − 0.412·53-s − 0.134·55-s + 1.17·59-s − 0.128·61-s + 0.496·65-s − 1.22·67-s − 0.712·71-s + 0.936·73-s + 0.113·77-s − 0.450·79-s − 0.329·83-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: Trivial
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 \( 1 + 3 T + p T^{2} \)
19 \( 1 + 7 T + p T^{2} \)
23 \( 1 - 9 T + p T^{2} \)
29 \( 1 - 3 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 + 3 T + p T^{2} \)
59 \( 1 - 9 T + p T^{2} \)
61 \( 1 + T + p T^{2} \)
67 \( 1 + 10 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 - 8 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 + 3 T + p T^{2} \)
89 \( 1 - 15 T + p T^{2} \)
97 \( 1 + T + p T^{2} \)
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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

−16.60207475974984, −15.73926696356621, −15.28801258303695, −14.70418333088265, −14.50560279041327, −13.55163017313769, −13.08193097582807, −12.38351556675140, −12.07132592892897, −11.20616096460669, −10.87415857060893, −10.27864938583851, −9.480234882214478, −8.805074798509303, −8.499626886309381, −7.629607928496493, −7.049067905119637, −6.594146897009253, −5.749810467834559, −4.777928988657147, −4.591465285659407, −3.739787638044648, −2.761499881133126, −2.203441083695600, −1.089761174837951, 0, 1.089761174837951, 2.203441083695600, 2.761499881133126, 3.739787638044648, 4.591465285659407, 4.777928988657147, 5.749810467834559, 6.594146897009253, 7.049067905119637, 7.629607928496493, 8.499626886309381, 8.805074798509303, 9.480234882214478, 10.27864938583851, 10.87415857060893, 11.20616096460669, 12.07132592892897, 12.38351556675140, 13.08193097582807, 13.55163017313769, 14.50560279041327, 14.70418333088265, 15.28801258303695, 15.73926696356621, 16.60207475974984

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