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Properties

Label 2-13860-1.1-c1-0-17
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 + 2·17-s − 6·19-s − 6·23-s + 25-s + 4·29-s + 6·31-s − 35-s + 2·37-s + 6·41-s + 4·43-s + 8·47-s + 49-s − 6·53-s + 55-s + 12·59-s + 14·61-s + 4·65-s + 4·67-s − 12·73-s − 77-s + 4·79-s − 2·85-s + 2·89-s + ⋯
L(s)  = 1  − 0.447·5-s + 0.377·7-s − 0.301·11-s − 1.10·13-s + 0.485·17-s − 1.37·19-s − 1.25·23-s + 1/5·25-s + 0.742·29-s + 1.07·31-s − 0.169·35-s + 0.328·37-s + 0.937·41-s + 0.609·43-s + 1.16·47-s + 1/7·49-s − 0.824·53-s + 0.134·55-s + 1.56·59-s + 1.79·61-s + 0.496·65-s + 0.488·67-s − 1.40·73-s − 0.113·77-s + 0.450·79-s − 0.216·85-s + 0.211·89-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 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 - 4 T + p T^{2} \)
31 \( 1 - 6 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 12 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 - 2 T + p T^{2} \)
97 \( 1 - 4 T + p T^{2} \)
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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

−16.27448641824749, −15.99300513420532, −15.26500939162776, −14.76535044831730, −14.27684191500097, −13.80671746085917, −12.82220427174016, −12.60373392899960, −11.83705652809940, −11.54346728934430, −10.58806046040970, −10.28878563431376, −9.638665913593808, −8.879295932308980, −8.097147995736404, −7.931223442271779, −7.099580941252777, −6.468690435982488, −5.724377904397412, −5.024343766540292, −4.291192521195051, −3.890953379774367, −2.615172700407878, −2.333469129703462, −1.056700411401312, 0, 1.056700411401312, 2.333469129703462, 2.615172700407878, 3.890953379774367, 4.291192521195051, 5.024343766540292, 5.724377904397412, 6.468690435982488, 7.099580941252777, 7.931223442271779, 8.097147995736404, 8.879295932308980, 9.638665913593808, 10.28878563431376, 10.58806046040970, 11.54346728934430, 11.83705652809940, 12.60373392899960, 12.82220427174016, 13.80671746085917, 14.27684191500097, 14.76535044831730, 15.26500939162776, 15.99300513420532, 16.27448641824749

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