Properties

Label 2-1386-1.1-c3-0-67
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 4·4-s + 17·5-s + 7·7-s − 8·8-s − 34·10-s + 11·11-s − 21·13-s − 14·14-s + 16·16-s + 104·17-s − 161·19-s + 68·20-s − 22·22-s − 194·23-s + 164·25-s + 42·26-s + 28·28-s − 9·29-s − 180·31-s − 32·32-s − 208·34-s + 119·35-s − 363·37-s + 322·38-s − 136·40-s + 108·41-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s + 1.52·5-s + 0.377·7-s − 0.353·8-s − 1.07·10-s + 0.301·11-s − 0.448·13-s − 0.267·14-s + 1/4·16-s + 1.48·17-s − 1.94·19-s + 0.760·20-s − 0.213·22-s − 1.75·23-s + 1.31·25-s + 0.316·26-s + 0.188·28-s − 0.0576·29-s − 1.04·31-s − 0.176·32-s − 1.04·34-s + 0.574·35-s − 1.61·37-s + 1.37·38-s − 0.537·40-s + 0.411·41-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: \(1\)
Selberg data: \((2,\ 1386,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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 \( 1 - 17 T + p^{3} T^{2} \)
13 \( 1 + 21 T + p^{3} T^{2} \)
17 \( 1 - 104 T + p^{3} T^{2} \)
19 \( 1 + 161 T + p^{3} T^{2} \)
23 \( 1 + 194 T + p^{3} T^{2} \)
29 \( 1 + 9 T + p^{3} T^{2} \)
31 \( 1 + 180 T + p^{3} T^{2} \)
37 \( 1 + 363 T + p^{3} T^{2} \)
41 \( 1 - 108 T + p^{3} T^{2} \)
43 \( 1 + 386 T + p^{3} T^{2} \)
47 \( 1 + 333 T + p^{3} T^{2} \)
53 \( 1 - 122 T + p^{3} T^{2} \)
59 \( 1 + 537 T + p^{3} T^{2} \)
61 \( 1 + 950 T + p^{3} T^{2} \)
67 \( 1 + 83 T + p^{3} T^{2} \)
71 \( 1 + 180 T + p^{3} T^{2} \)
73 \( 1 - 177 T + p^{3} T^{2} \)
79 \( 1 + 220 T + p^{3} T^{2} \)
83 \( 1 + 1112 T + p^{3} T^{2} \)
89 \( 1 - 394 T + p^{3} T^{2} \)
97 \( 1 - 826 T + p^{3} T^{2} \)
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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

−8.868106862537499909814833785526, −8.161826612555130518738309954568, −7.22840220007416147632141059083, −6.21878468208738307763427082733, −5.77770843850026013640364384836, −4.70602494483231923046374680007, −3.35795312022273318179532010056, −1.99565030426506773879638303031, −1.66621217915994825606121072802, 0, 1.66621217915994825606121072802, 1.99565030426506773879638303031, 3.35795312022273318179532010056, 4.70602494483231923046374680007, 5.77770843850026013640364384836, 6.21878468208738307763427082733, 7.22840220007416147632141059083, 8.161826612555130518738309954568, 8.868106862537499909814833785526

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