Properties

Label 2-1386-1.1-c3-0-42
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 − 18·5-s + 7·7-s − 8·8-s + 36·10-s + 11·11-s + 56·13-s − 14·14-s + 16·16-s − 36·17-s − 28·19-s − 72·20-s − 22·22-s − 180·23-s + 199·25-s − 112·26-s + 28·28-s + 54·29-s − 334·31-s − 32·32-s + 72·34-s − 126·35-s + 386·37-s + 56·38-s + 144·40-s + 444·41-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 1.60·5-s + 0.377·7-s − 0.353·8-s + 1.13·10-s + 0.301·11-s + 1.19·13-s − 0.267·14-s + 1/4·16-s − 0.513·17-s − 0.338·19-s − 0.804·20-s − 0.213·22-s − 1.63·23-s + 1.59·25-s − 0.844·26-s + 0.188·28-s + 0.345·29-s − 1.93·31-s − 0.176·32-s + 0.363·34-s − 0.608·35-s + 1.71·37-s + 0.239·38-s + 0.569·40-s + 1.69·41-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 \( 1 + 18 T + p^{3} T^{2} \)
13 \( 1 - 56 T + p^{3} T^{2} \)
17 \( 1 + 36 T + p^{3} T^{2} \)
19 \( 1 + 28 T + p^{3} T^{2} \)
23 \( 1 + 180 T + p^{3} T^{2} \)
29 \( 1 - 54 T + p^{3} T^{2} \)
31 \( 1 + 334 T + p^{3} T^{2} \)
37 \( 1 - 386 T + p^{3} T^{2} \)
41 \( 1 - 444 T + p^{3} T^{2} \)
43 \( 1 + 316 T + p^{3} T^{2} \)
47 \( 1 - 402 T + p^{3} T^{2} \)
53 \( 1 - 486 T + p^{3} T^{2} \)
59 \( 1 - 282 T + p^{3} T^{2} \)
61 \( 1 - 380 T + p^{3} T^{2} \)
67 \( 1 - 176 T + p^{3} T^{2} \)
71 \( 1 - 324 T + p^{3} T^{2} \)
73 \( 1 - 800 T + p^{3} T^{2} \)
79 \( 1 + 1144 T + p^{3} T^{2} \)
83 \( 1 + 468 T + p^{3} T^{2} \)
89 \( 1 - 870 T + p^{3} T^{2} \)
97 \( 1 + 1330 T + p^{3} T^{2} \)
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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

−8.601958322180537194457897430896, −8.093757181429723510854923906201, −7.43940937037103612952648823025, −6.56181679555120029678857060256, −5.60202846449636198930935151356, −4.06505328134045114802122915177, −3.89503016056136020739562413085, −2.39520235077224935073664593844, −1.07223753027673897970923987892, 0, 1.07223753027673897970923987892, 2.39520235077224935073664593844, 3.89503016056136020739562413085, 4.06505328134045114802122915177, 5.60202846449636198930935151356, 6.56181679555120029678857060256, 7.43940937037103612952648823025, 8.093757181429723510854923906201, 8.601958322180537194457897430896

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