Properties

Label 2-13860-1.1-c1-0-0
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 − 6·13-s − 2·17-s − 6·23-s + 25-s − 6·29-s − 2·31-s − 35-s + 10·37-s + 8·41-s − 8·43-s − 4·47-s + 49-s − 6·53-s − 55-s + 6·59-s − 8·61-s − 6·65-s + 14·67-s + 8·71-s − 2·73-s + 77-s − 16·79-s − 12·83-s − 2·85-s + ⋯
L(s)  = 1  + 0.447·5-s − 0.377·7-s − 0.301·11-s − 1.66·13-s − 0.485·17-s − 1.25·23-s + 1/5·25-s − 1.11·29-s − 0.359·31-s − 0.169·35-s + 1.64·37-s + 1.24·41-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 − 0.744·65-s + 1.71·67-s + 0.949·71-s − 0.234·73-s + 0.113·77-s − 1.80·79-s − 1.31·83-s − 0.216·85-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: \(0\)
Selberg data: \((2,\ 13860,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.156617055\)
\(L(\frac12)\) \(\approx\) \(1.156617055\)
\(L(\frac{3}{2})\) 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 \( 1 + 6 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 - 8 T + p T^{2} \)
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 \( 1 - 14 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 18 T + p T^{2} \)
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

−16.17431361150645, −15.65302472261497, −14.96617317016404, −14.43924234512184, −14.10123585218268, −13.11023401829073, −12.93914438622141, −12.34711976184431, −11.54056769488189, −11.18459222734753, −10.21230837121173, −9.876470651058308, −9.423394709190449, −8.740213223208287, −7.800936530150139, −7.537442036673368, −6.688805620878227, −6.095735428669098, −5.441469537007842, −4.751570789634387, −4.107196903298041, −3.172961996262869, −2.380296334492648, −1.874938845780561, −0.4473188442077274, 0.4473188442077274, 1.874938845780561, 2.380296334492648, 3.172961996262869, 4.107196903298041, 4.751570789634387, 5.441469537007842, 6.095735428669098, 6.688805620878227, 7.537442036673368, 7.800936530150139, 8.740213223208287, 9.423394709190449, 9.876470651058308, 10.21230837121173, 11.18459222734753, 11.54056769488189, 12.34711976184431, 12.93914438622141, 13.11023401829073, 14.10123585218268, 14.43924234512184, 14.96617317016404, 15.65302472261497, 16.17431361150645

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