Properties

Label 2-1386-1.1-c1-0-11
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s + 4·5-s + 7-s − 8-s − 4·10-s + 11-s − 6·13-s − 14-s + 16-s + 4·17-s − 2·19-s + 4·20-s − 22-s + 8·23-s + 11·25-s + 6·26-s + 28-s + 6·29-s + 6·31-s − 32-s − 4·34-s + 4·35-s − 6·37-s + 2·38-s − 4·40-s − 12·41-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s + 1.78·5-s + 0.377·7-s − 0.353·8-s − 1.26·10-s + 0.301·11-s − 1.66·13-s − 0.267·14-s + 1/4·16-s + 0.970·17-s − 0.458·19-s + 0.894·20-s − 0.213·22-s + 1.66·23-s + 11/5·25-s + 1.17·26-s + 0.188·28-s + 1.11·29-s + 1.07·31-s − 0.176·32-s − 0.685·34-s + 0.676·35-s − 0.986·37-s + 0.324·38-s − 0.632·40-s − 1.87·41-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: \(0\)
Selberg data: \((2,\ 1386,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.780456472\)
\(L(\frac12)\) \(\approx\) \(1.780456472\)
\(L(\frac{3}{2})\) 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 \( 1 - 4 T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 6 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 12 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 - 14 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−9.739712159295331966490625255940, −8.917387992761630968547443589168, −8.140353505930131920153142154645, −6.96548548882838290586125297122, −6.54704897497816919862448175054, −5.33392541266418642466287088585, −4.91457124178222642670967255706, −3.01743557007021262394668003027, −2.19563344920726121984618617678, −1.16576401497843678501279592702, 1.16576401497843678501279592702, 2.19563344920726121984618617678, 3.01743557007021262394668003027, 4.91457124178222642670967255706, 5.33392541266418642466287088585, 6.54704897497816919862448175054, 6.96548548882838290586125297122, 8.140353505930131920153142154645, 8.917387992761630968547443589168, 9.739712159295331966490625255940

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