Properties

Label 2-13860-1.1-c1-0-20
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 + 2·17-s − 6·19-s − 8·23-s + 25-s + 4·31-s − 35-s − 2·37-s − 8·41-s + 8·43-s + 12·47-s + 49-s + 10·53-s + 55-s + 12·59-s − 2·61-s − 4·67-s + 8·71-s − 12·73-s − 77-s − 10·79-s + 2·83-s + 2·85-s + 6·89-s − 6·95-s + ⋯
L(s)  = 1  + 0.447·5-s − 0.377·7-s + 0.301·11-s + 0.485·17-s − 1.37·19-s − 1.66·23-s + 1/5·25-s + 0.718·31-s − 0.169·35-s − 0.328·37-s − 1.24·41-s + 1.21·43-s + 1.75·47-s + 1/7·49-s + 1.37·53-s + 0.134·55-s + 1.56·59-s − 0.256·61-s − 0.488·67-s + 0.949·71-s − 1.40·73-s − 0.113·77-s − 1.12·79-s + 0.219·83-s + 0.216·85-s + 0.635·89-s − 0.615·95-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 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 8 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 12 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 - 2 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 18 T + p T^{2} \)
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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

−16.45104284688500, −15.97054737373478, −15.25403164423312, −14.83077852914301, −14.06598345093488, −13.75445676748822, −13.11809360371039, −12.39172141931366, −12.09709218185192, −11.39678525370189, −10.51880093256021, −10.21767046935531, −9.685230221255115, −8.821331348343823, −8.512689625826605, −7.692521844668009, −6.993634090274054, −6.340448108172127, −5.850384783389313, −5.207008190036075, −4.108295659073683, −3.917576438455564, −2.714954032501265, −2.170804326705937, −1.187004184304927, 0, 1.187004184304927, 2.170804326705937, 2.714954032501265, 3.917576438455564, 4.108295659073683, 5.207008190036075, 5.850384783389313, 6.340448108172127, 6.993634090274054, 7.692521844668009, 8.512689625826605, 8.821331348343823, 9.685230221255115, 10.21767046935531, 10.51880093256021, 11.39678525370189, 12.09709218185192, 12.39172141931366, 13.11809360371039, 13.75445676748822, 14.06598345093488, 14.83077852914301, 15.25403164423312, 15.97054737373478, 16.45104284688500

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