Properties

Label 2-1386-1.1-c1-0-14
Degree $2$
Conductor $1386$
Sign $1$
Analytic cond. $11.0672$
Root an. cond. $3.32675$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s + 3.16·5-s − 7-s + 8-s + 3.16·10-s − 11-s + 2·13-s − 14-s + 16-s + 0.837·17-s − 1.16·19-s + 3.16·20-s − 22-s + 6.32·23-s + 5.00·25-s + 2·26-s − 28-s + 4·29-s + 2.83·31-s + 32-s + 0.837·34-s − 3.16·35-s − 4.32·37-s − 1.16·38-s + 3.16·40-s − 0.837·41-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s + 1.41·5-s − 0.377·7-s + 0.353·8-s + 1.00·10-s − 0.301·11-s + 0.554·13-s − 0.267·14-s + 0.250·16-s + 0.203·17-s − 0.266·19-s + 0.707·20-s − 0.213·22-s + 1.31·23-s + 1.00·25-s + 0.392·26-s − 0.188·28-s + 0.742·29-s + 0.509·31-s + 0.176·32-s + 0.143·34-s − 0.534·35-s − 0.710·37-s − 0.188·38-s + 0.500·40-s − 0.130·41-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: $1$
Analytic conductor: \(11.0672\)
Root analytic conductor: \(3.32675\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1386} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1386,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.326905963\)
\(L(\frac12)\) \(\approx\) \(3.326905963\)
\(L(\frac{3}{2})\) 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 \( 1 - 3.16T + 5T^{2} \)
13 \( 1 - 2T + 13T^{2} \)
17 \( 1 - 0.837T + 17T^{2} \)
19 \( 1 + 1.16T + 19T^{2} \)
23 \( 1 - 6.32T + 23T^{2} \)
29 \( 1 - 4T + 29T^{2} \)
31 \( 1 - 2.83T + 31T^{2} \)
37 \( 1 + 4.32T + 37T^{2} \)
41 \( 1 + 0.837T + 41T^{2} \)
43 \( 1 - 6.32T + 43T^{2} \)
47 \( 1 + 7.48T + 47T^{2} \)
53 \( 1 + 8.32T + 53T^{2} \)
59 \( 1 + 14.3T + 59T^{2} \)
61 \( 1 - 10T + 61T^{2} \)
67 \( 1 + 8.32T + 67T^{2} \)
71 \( 1 + 8T + 71T^{2} \)
73 \( 1 - 13.4T + 73T^{2} \)
79 \( 1 - 8.32T + 79T^{2} \)
83 \( 1 - 1.16T + 83T^{2} \)
89 \( 1 - 3.67T + 89T^{2} \)
97 \( 1 - 14.6T + 97T^{2} \)
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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

−9.616630476242853064208855805351, −8.937713895707074353855091783864, −7.895399363381712292026469778154, −6.72897918412488033891411690667, −6.25775157059139485414936165440, −5.41651533802523318220085879605, −4.68428278788750116352676484791, −3.36080542663436974401221328048, −2.52686133606690634051205734907, −1.36980157415612259587668488149, 1.36980157415612259587668488149, 2.52686133606690634051205734907, 3.36080542663436974401221328048, 4.68428278788750116352676484791, 5.41651533802523318220085879605, 6.25775157059139485414936165440, 6.72897918412488033891411690667, 7.895399363381712292026469778154, 8.937713895707074353855091783864, 9.616630476242853064208855805351

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