Properties

Label 2-13860-1.1-c1-0-16
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 + 6·17-s − 6·19-s + 4·23-s + 25-s − 8·29-s − 8·31-s + 35-s + 6·37-s + 4·41-s + 12·47-s + 49-s + 6·53-s + 55-s + 4·59-s − 10·61-s + 4·67-s − 8·71-s − 4·73-s + 77-s + 2·79-s + 18·83-s − 6·85-s − 6·89-s + 6·95-s + ⋯
L(s)  = 1  − 0.447·5-s − 0.377·7-s − 0.301·11-s + 1.45·17-s − 1.37·19-s + 0.834·23-s + 1/5·25-s − 1.48·29-s − 1.43·31-s + 0.169·35-s + 0.986·37-s + 0.624·41-s + 1.75·47-s + 1/7·49-s + 0.824·53-s + 0.134·55-s + 0.520·59-s − 1.28·61-s + 0.488·67-s − 0.949·71-s − 0.468·73-s + 0.113·77-s + 0.225·79-s + 1.97·83-s − 0.650·85-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 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 8 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 - 4 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 - 2 T + p T^{2} \)
83 \( 1 - 18 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
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

−16.52296435788981, −15.99772418188439, −15.09547457143634, −14.90869549161202, −14.41290904905033, −13.47993148746966, −13.05958812899056, −12.47898433800416, −12.09481327641409, −11.20883966492993, −10.79308681814232, −10.28623852702667, −9.405029065202531, −9.071912480529059, −8.303072628603439, −7.502948684465963, −7.347961618610840, −6.369528011358267, −5.710472383720719, −5.207875420119959, −4.184386843683818, −3.749644327231435, −2.925648813752947, −2.144015782402475, −1.056444203329655, 0, 1.056444203329655, 2.144015782402475, 2.925648813752947, 3.749644327231435, 4.184386843683818, 5.207875420119959, 5.710472383720719, 6.369528011358267, 7.347961618610840, 7.502948684465963, 8.303072628603439, 9.071912480529059, 9.405029065202531, 10.28623852702667, 10.79308681814232, 11.20883966492993, 12.09481327641409, 12.47898433800416, 13.05958812899056, 13.47993148746966, 14.41290904905033, 14.90869549161202, 15.09547457143634, 15.99772418188439, 16.52296435788981

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