Properties

Label 2-13860-1.1-c1-0-2
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 − 6·17-s − 8·23-s + 25-s − 2·29-s − 8·31-s + 35-s + 6·37-s + 10·41-s + 49-s − 6·53-s + 55-s − 2·59-s + 8·61-s − 6·65-s − 2·67-s − 8·71-s + 14·73-s + 77-s − 10·79-s − 6·83-s + 6·85-s + 6·89-s − 6·91-s + ⋯
L(s)  = 1  − 0.447·5-s − 0.377·7-s − 0.301·11-s + 1.66·13-s − 1.45·17-s − 1.66·23-s + 1/5·25-s − 0.371·29-s − 1.43·31-s + 0.169·35-s + 0.986·37-s + 1.56·41-s + 1/7·49-s − 0.824·53-s + 0.134·55-s − 0.260·59-s + 1.02·61-s − 0.744·65-s − 0.244·67-s − 0.949·71-s + 1.63·73-s + 0.113·77-s − 1.12·79-s − 0.658·83-s + 0.650·85-s + 0.635·89-s − 0.628·91-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: $\chi_{13860} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 13860,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.258030623\)
\(L(\frac12)\) \(\approx\) \(1.258030623\)
\(L(\frac{3}{2})\) 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 \( 1 + 6 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 2 T + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
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

−16.10748766002053, −15.78698170951246, −15.17035881801274, −14.46293251021058, −13.91703232461507, −13.16627518522553, −13.00259665660437, −12.26748602805730, −11.46543374845849, −11.04970310013598, −10.68592379563984, −9.791713084048767, −9.201903058465322, −8.634880354528476, −8.052941205290186, −7.452766428815295, −6.672143855479476, −6.055969602999843, −5.643774208837058, −4.497069864959352, −4.018624426441504, −3.434586276739137, −2.456403759570575, −1.679018226638758, −0.4892228966110833, 0.4892228966110833, 1.679018226638758, 2.456403759570575, 3.434586276739137, 4.018624426441504, 4.497069864959352, 5.643774208837058, 6.055969602999843, 6.672143855479476, 7.452766428815295, 8.052941205290186, 8.634880354528476, 9.201903058465322, 9.791713084048767, 10.68592379563984, 11.04970310013598, 11.46543374845849, 12.26748602805730, 13.00259665660437, 13.16627518522553, 13.91703232461507, 14.46293251021058, 15.17035881801274, 15.78698170951246, 16.10748766002053

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