Properties

Label 2-1386-1.1-c1-0-24
Degree $2$
Conductor $1386$
Sign $-1$
Analytic cond. $11.0672$
Root an. cond. $3.32675$
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  + 2-s + 4-s − 2·5-s − 7-s + 8-s − 2·10-s + 11-s − 4·13-s − 14-s + 16-s − 2·17-s − 6·19-s − 2·20-s + 22-s + 2·23-s − 25-s − 4·26-s − 28-s + 2·29-s − 8·31-s + 32-s − 2·34-s + 2·35-s − 2·37-s − 6·38-s − 2·40-s + 2·41-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 0.894·5-s − 0.377·7-s + 0.353·8-s − 0.632·10-s + 0.301·11-s − 1.10·13-s − 0.267·14-s + 1/4·16-s − 0.485·17-s − 1.37·19-s − 0.447·20-s + 0.213·22-s + 0.417·23-s − 1/5·25-s − 0.784·26-s − 0.188·28-s + 0.371·29-s − 1.43·31-s + 0.176·32-s − 0.342·34-s + 0.338·35-s − 0.328·37-s − 0.973·38-s − 0.316·40-s + 0.312·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1386\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 11\)
Sign: $-1$
Analytic conductor: \(11.0672\)
Root analytic conductor: \(3.32675\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1386} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1386,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
3 \( 1 \)
7 \( 1 + T \)
11 \( 1 - T \)
good5 \( 1 + 2 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 - 2 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 + 2 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 4 T + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 + 10 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 12 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 12 T + p T^{2} \)
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

−9.134494299607741468141713783567, −8.272162289910099551122345576861, −7.34142071798424504356409604185, −6.78155434392206371572794646550, −5.81531940011466945768947277432, −4.73109850526366519062683146302, −4.09136916114338661687753966711, −3.14508766922258374979544108268, −2.02258435897521586180281099584, 0, 2.02258435897521586180281099584, 3.14508766922258374979544108268, 4.09136916114338661687753966711, 4.73109850526366519062683146302, 5.81531940011466945768947277432, 6.78155434392206371572794646550, 7.34142071798424504356409604185, 8.272162289910099551122345576861, 9.134494299607741468141713783567

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