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Properties

Label 2-13860-1.1-c1-0-3
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5-s + 7-s + 11-s + 2·13-s − 6·17-s − 4·19-s + 25-s − 6·29-s + 8·31-s − 35-s − 10·37-s + 6·41-s + 8·43-s + 49-s − 6·53-s − 55-s + 6·59-s − 4·61-s − 2·65-s + 14·67-s + 2·73-s + 77-s − 10·79-s − 6·83-s + 6·85-s − 18·89-s + 2·91-s + ⋯
L(s)  = 1  − 0.447·5-s + 0.377·7-s + 0.301·11-s + 0.554·13-s − 1.45·17-s − 0.917·19-s + 1/5·25-s − 1.11·29-s + 1.43·31-s − 0.169·35-s − 1.64·37-s + 0.937·41-s + 1.21·43-s + 1/7·49-s − 0.824·53-s − 0.134·55-s + 0.781·59-s − 0.512·61-s − 0.248·65-s + 1.71·67-s + 0.234·73-s + 0.113·77-s − 1.12·79-s − 0.658·83-s + 0.650·85-s − 1.90·89-s + 0.209·91-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: \(0\)
Selberg data: \((2,\ 13860,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.620403386\)
\(L(\frac12)\) \(\approx\) \(1.620403386\)
\(L(\frac{3}{2})\) 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 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 + 4 T + p T^{2} \)
67 \( 1 - 14 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 + 18 T + p T^{2} \)
97 \( 1 - 8 T + p T^{2} \)
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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

−15.92148431336593, −15.63477320382768, −15.21409286816112, −14.41162951708633, −14.02777710542832, −13.33802894849433, −12.75680998487314, −12.29609823028762, −11.39977031631442, −11.18326275978787, −10.62804498136242, −9.871449941252015, −9.094844685452564, −8.603861986628944, −8.177782768643918, −7.320592262022005, −6.778275317032057, −6.154585463307482, −5.442922073722890, −4.462448437351920, −4.220045312588592, −3.357236429940214, −2.401156612447801, −1.696762735425288, −0.5601416009091844, 0.5601416009091844, 1.696762735425288, 2.401156612447801, 3.357236429940214, 4.220045312588592, 4.462448437351920, 5.442922073722890, 6.154585463307482, 6.778275317032057, 7.320592262022005, 8.177782768643918, 8.603861986628944, 9.094844685452564, 9.871449941252015, 10.62804498136242, 11.18326275978787, 11.39977031631442, 12.29609823028762, 12.75680998487314, 13.33802894849433, 14.02777710542832, 14.41162951708633, 15.21409286816112, 15.63477320382768, 15.92148431336593

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