Properties

Label 2-13860-1.1-c1-0-21
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 − 3·17-s − 19-s + 7·23-s + 25-s − 5·29-s − 6·31-s − 35-s + 8·37-s + 2·41-s − 7·43-s + 2·47-s + 49-s − 5·53-s + 55-s + 7·59-s − 7·61-s − 14·67-s − 2·71-s − 12·73-s − 77-s − 3·83-s − 3·85-s + 89-s − 95-s + ⋯
L(s)  = 1  + 0.447·5-s − 0.377·7-s + 0.301·11-s − 0.727·17-s − 0.229·19-s + 1.45·23-s + 1/5·25-s − 0.928·29-s − 1.07·31-s − 0.169·35-s + 1.31·37-s + 0.312·41-s − 1.06·43-s + 0.291·47-s + 1/7·49-s − 0.686·53-s + 0.134·55-s + 0.911·59-s − 0.896·61-s − 1.71·67-s − 0.237·71-s − 1.40·73-s − 0.113·77-s − 0.329·83-s − 0.325·85-s + 0.105·89-s − 0.102·95-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: \(1\)
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 \( 1 + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 + T + p T^{2} \)
23 \( 1 - 7 T + p T^{2} \)
29 \( 1 + 5 T + p T^{2} \)
31 \( 1 + 6 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 7 T + p T^{2} \)
47 \( 1 - 2 T + p T^{2} \)
53 \( 1 + 5 T + p T^{2} \)
59 \( 1 - 7 T + p T^{2} \)
61 \( 1 + 7 T + p T^{2} \)
67 \( 1 + 14 T + p T^{2} \)
71 \( 1 + 2 T + p T^{2} \)
73 \( 1 + 12 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 3 T + p T^{2} \)
89 \( 1 - T + p T^{2} \)
97 \( 1 - 7 T + p T^{2} \)
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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

−16.56787719314794, −15.93498772429573, −15.22003074335095, −14.76514682371691, −14.35504495464692, −13.36252954194227, −13.19092890170413, −12.71245801731836, −11.88023899497104, −11.25467885913277, −10.83491692115675, −10.13549781148491, −9.460025970455768, −9.010380367930908, −8.551220843632686, −7.461860955220676, −7.181710089818785, −6.284878268510926, −5.925799549152605, −5.036774322691365, −4.450855068243863, −3.590432093357313, −2.892783530658035, −2.068293352275671, −1.214686226481560, 0, 1.214686226481560, 2.068293352275671, 2.892783530658035, 3.590432093357313, 4.450855068243863, 5.036774322691365, 5.925799549152605, 6.284878268510926, 7.181710089818785, 7.461860955220676, 8.551220843632686, 9.010380367930908, 9.460025970455768, 10.13549781148491, 10.83491692115675, 11.25467885913277, 11.88023899497104, 12.71245801731836, 13.19092890170413, 13.36252954194227, 14.35504495464692, 14.76514682371691, 15.22003074335095, 15.93498772429573, 16.56787719314794

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