Properties

Label 2-1386-1.1-c3-0-50
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 − 3·5-s + 7·7-s − 8·8-s + 6·10-s + 11·11-s + 41·13-s − 14·14-s + 16·16-s − 6·17-s − 43·19-s − 12·20-s − 22·22-s − 120·23-s − 116·25-s − 82·26-s + 28·28-s − 111·29-s + 266·31-s − 32·32-s + 12·34-s − 21·35-s − 79·37-s + 86·38-s + 24·40-s − 216·41-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.268·5-s + 0.377·7-s − 0.353·8-s + 0.189·10-s + 0.301·11-s + 0.874·13-s − 0.267·14-s + 1/4·16-s − 0.0856·17-s − 0.519·19-s − 0.134·20-s − 0.213·22-s − 1.08·23-s − 0.927·25-s − 0.618·26-s + 0.188·28-s − 0.710·29-s + 1.54·31-s − 0.176·32-s + 0.0605·34-s − 0.101·35-s − 0.351·37-s + 0.367·38-s + 0.0948·40-s − 0.822·41-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 \( 1 + 3 T + p^{3} T^{2} \)
13 \( 1 - 41 T + p^{3} T^{2} \)
17 \( 1 + 6 T + p^{3} T^{2} \)
19 \( 1 + 43 T + p^{3} T^{2} \)
23 \( 1 + 120 T + p^{3} T^{2} \)
29 \( 1 + 111 T + p^{3} T^{2} \)
31 \( 1 - 266 T + p^{3} T^{2} \)
37 \( 1 + 79 T + p^{3} T^{2} \)
41 \( 1 + 216 T + p^{3} T^{2} \)
43 \( 1 - 284 T + p^{3} T^{2} \)
47 \( 1 + 213 T + p^{3} T^{2} \)
53 \( 1 - 216 T + p^{3} T^{2} \)
59 \( 1 + 393 T + p^{3} T^{2} \)
61 \( 1 - 350 T + p^{3} T^{2} \)
67 \( 1 - 821 T + p^{3} T^{2} \)
71 \( 1 - 264 T + p^{3} T^{2} \)
73 \( 1 + 865 T + p^{3} T^{2} \)
79 \( 1 + 484 T + p^{3} T^{2} \)
83 \( 1 + 1158 T + p^{3} T^{2} \)
89 \( 1 + 330 T + p^{3} T^{2} \)
97 \( 1 - 980 T + p^{3} T^{2} \)
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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

−8.592581425148084401920757488349, −8.222552021114660242433348568264, −7.33183622986043499678808230232, −6.39964827659824922233042458134, −5.70222973852395637493389827930, −4.40168393276223592932355444060, −3.58788784566443930872412043770, −2.27163114157772339316552266883, −1.28147247346995776795387367784, 0, 1.28147247346995776795387367784, 2.27163114157772339316552266883, 3.58788784566443930872412043770, 4.40168393276223592932355444060, 5.70222973852395637493389827930, 6.39964827659824922233042458134, 7.33183622986043499678808230232, 8.222552021114660242433348568264, 8.592581425148084401920757488349

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