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Properties

Label 2-13860-1.1-c1-0-8
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 − 2·17-s + 6·23-s + 25-s + 2·29-s − 6·31-s + 35-s + 2·37-s − 8·43-s + 4·47-s + 49-s − 6·53-s + 55-s − 6·59-s + 8·61-s + 2·65-s + 10·67-s + 6·73-s + 77-s + 16·79-s + 12·83-s − 2·85-s + 10·89-s + 2·91-s + ⋯
L(s)  = 1  + 0.447·5-s + 0.377·7-s + 0.301·11-s + 0.554·13-s − 0.485·17-s + 1.25·23-s + 1/5·25-s + 0.371·29-s − 1.07·31-s + 0.169·35-s + 0.328·37-s − 1.21·43-s + 0.583·47-s + 1/7·49-s − 0.824·53-s + 0.134·55-s − 0.781·59-s + 1.02·61-s + 0.248·65-s + 1.22·67-s + 0.702·73-s + 0.113·77-s + 1.80·79-s + 1.31·83-s − 0.216·85-s + 1.05·89-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

\(L(1)\) \(\approx\) \(2.706558650\)
\(L(\frac12)\) \(\approx\) \(2.706558650\)
\(L(\frac{3}{2})\) 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 \( 1 + 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 6 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 - 10 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−16.28920014484238, −15.47685431620820, −15.09126066061585, −14.46058825744793, −13.94256952564496, −13.35176609975188, −12.88271652049996, −12.27440763764477, −11.50764636939338, −11.03188313428159, −10.59922088269786, −9.798932016917731, −9.173336310509576, −8.769987471454095, −8.051091821758488, −7.384034943880288, −6.591423022192550, −6.265765509794863, −5.225551573956210, −4.952814295768977, −3.956552055396029, −3.332341518931595, −2.392688944285375, −1.645944291716248, −0.7556474447211747, 0.7556474447211747, 1.645944291716248, 2.392688944285375, 3.332341518931595, 3.956552055396029, 4.952814295768977, 5.225551573956210, 6.265765509794863, 6.591423022192550, 7.384034943880288, 8.051091821758488, 8.769987471454095, 9.173336310509576, 9.798932016917731, 10.59922088269786, 11.03188313428159, 11.50764636939338, 12.27440763764477, 12.88271652049996, 13.35176609975188, 13.94256952564496, 14.46058825744793, 15.09126066061585, 15.47685431620820, 16.28920014484238

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