Properties

Label 2-1386-1.1-c3-0-4
Degree $2$
Conductor $1386$
Sign $1$
Analytic cond. $81.7766$
Root an. cond. $9.04304$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 4·4-s − 5-s − 7·7-s − 8·8-s + 2·10-s + 11·11-s − 43·13-s + 14·14-s + 16·16-s − 100·17-s − 87·19-s − 4·20-s − 22·22-s + 58·23-s − 124·25-s + 86·26-s − 28·28-s + 223·29-s + 88·31-s − 32·32-s + 200·34-s + 7·35-s + 37·37-s + 174·38-s + 8·40-s − 128·41-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.0894·5-s − 0.377·7-s − 0.353·8-s + 0.0632·10-s + 0.301·11-s − 0.917·13-s + 0.267·14-s + 1/4·16-s − 1.42·17-s − 1.05·19-s − 0.0447·20-s − 0.213·22-s + 0.525·23-s − 0.991·25-s + 0.648·26-s − 0.188·28-s + 1.42·29-s + 0.509·31-s − 0.176·32-s + 1.00·34-s + 0.0338·35-s + 0.164·37-s + 0.742·38-s + 0.0316·40-s − 0.487·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1386\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 11\)
Sign: $1$
Analytic conductor: \(81.7766\)
Root analytic conductor: \(9.04304\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1386} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1386,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(0.8296627562\)
\(L(\frac12)\) \(\approx\) \(0.8296627562\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + p T \)
3 \( 1 \)
7 \( 1 + p T \)
11 \( 1 - p T \)
good5 \( 1 + T + p^{3} T^{2} \)
13 \( 1 + 43 T + p^{3} T^{2} \)
17 \( 1 + 100 T + p^{3} T^{2} \)
19 \( 1 + 87 T + p^{3} T^{2} \)
23 \( 1 - 58 T + p^{3} T^{2} \)
29 \( 1 - 223 T + p^{3} T^{2} \)
31 \( 1 - 88 T + p^{3} T^{2} \)
37 \( 1 - p T + p^{3} T^{2} \)
41 \( 1 + 128 T + p^{3} T^{2} \)
43 \( 1 + 458 T + p^{3} T^{2} \)
47 \( 1 - 341 T + p^{3} T^{2} \)
53 \( 1 - 342 T + p^{3} T^{2} \)
59 \( 1 - 105 T + p^{3} T^{2} \)
61 \( 1 - 190 T + p^{3} T^{2} \)
67 \( 1 + 579 T + p^{3} T^{2} \)
71 \( 1 + 128 T + p^{3} T^{2} \)
73 \( 1 + 161 T + p^{3} T^{2} \)
79 \( 1 + 396 T + p^{3} T^{2} \)
83 \( 1 - 420 T + p^{3} T^{2} \)
89 \( 1 - 798 T + p^{3} T^{2} \)
97 \( 1 - 1414 T + p^{3} T^{2} \)
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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

−9.052905243627381320898242357349, −8.606941269522839390263439703792, −7.63224114468761389804303628415, −6.75112532867875006946541864100, −6.27316328537433767470638673823, −4.96018551802514596430532750286, −4.07723964136555717475109753825, −2.79690840261545848484801412158, −1.94912373014348482263167980671, −0.48259118971023212897761976458, 0.48259118971023212897761976458, 1.94912373014348482263167980671, 2.79690840261545848484801412158, 4.07723964136555717475109753825, 4.96018551802514596430532750286, 6.27316328537433767470638673823, 6.75112532867875006946541864100, 7.63224114468761389804303628415, 8.606941269522839390263439703792, 9.052905243627381320898242357349

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