Properties

Degree $2$
Conductor $1386$
Sign $1$
Motivic weight $1$
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 + 2·13-s − 14-s + 16-s + 4·17-s − 6·19-s + 4·20-s + 22-s − 4·23-s + 11·25-s + 2·26-s − 28-s + 2·29-s − 2·31-s + 32-s + 4·34-s − 4·35-s + 10·37-s − 6·38-s + 4·40-s − 4·41-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 + 0.554·13-s − 0.267·14-s + 1/4·16-s + 0.970·17-s − 1.37·19-s + 0.894·20-s + 0.213·22-s − 0.834·23-s + 11/5·25-s + 0.392·26-s − 0.188·28-s + 0.371·29-s − 0.359·31-s + 0.176·32-s + 0.685·34-s − 0.676·35-s + 1.64·37-s − 0.973·38-s + 0.632·40-s − 0.624·41-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: $1$
Motivic weight: \(1\)
Character: $\chi_{1386} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1386,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.553477576\)
\(L(\frac12)\) \(\approx\) \(3.553477576\)
\(L(\frac{3}{2})\) 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 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + 4 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + 2 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
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

−9.759674701374569314122222212467, −8.907037361397106316846384330744, −7.951642576699392563387512465158, −6.62445475159785453495873048483, −6.22501908736119603038057345164, −5.57547584921431243576094737475, −4.59118714632791292827191258849, −3.44639178218693714912433679777, −2.38911105843277843359160362418, −1.46845333940904453855522694316, 1.46845333940904453855522694316, 2.38911105843277843359160362418, 3.44639178218693714912433679777, 4.59118714632791292827191258849, 5.57547584921431243576094737475, 6.22501908736119603038057345164, 6.62445475159785453495873048483, 7.951642576699392563387512465158, 8.907037361397106316846384330744, 9.759674701374569314122222212467

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