Properties

Label 2-1386-1.1-c3-0-24
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 − 2·5-s − 7·7-s + 8·8-s − 4·10-s − 11·11-s + 26·13-s − 14·14-s + 16·16-s + 46·17-s − 48·19-s − 8·20-s − 22·22-s + 128·23-s − 121·25-s + 52·26-s − 28·28-s + 146·29-s − 128·31-s + 32·32-s + 92·34-s + 14·35-s − 26·37-s − 96·38-s − 16·40-s − 10·41-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 0.178·5-s − 0.377·7-s + 0.353·8-s − 0.126·10-s − 0.301·11-s + 0.554·13-s − 0.267·14-s + 1/4·16-s + 0.656·17-s − 0.579·19-s − 0.0894·20-s − 0.213·22-s + 1.16·23-s − 0.967·25-s + 0.392·26-s − 0.188·28-s + 0.934·29-s − 0.741·31-s + 0.176·32-s + 0.464·34-s + 0.0676·35-s − 0.115·37-s − 0.409·38-s − 0.0632·40-s − 0.0380·41-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

\(L(2)\) \(\approx\) \(3.209818819\)
\(L(\frac12)\) \(\approx\) \(3.209818819\)
\(L(\frac{5}{2})\) 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 \( 1 + 2 T + p^{3} T^{2} \)
13 \( 1 - 2 p T + p^{3} T^{2} \)
17 \( 1 - 46 T + p^{3} T^{2} \)
19 \( 1 + 48 T + p^{3} T^{2} \)
23 \( 1 - 128 T + p^{3} T^{2} \)
29 \( 1 - 146 T + p^{3} T^{2} \)
31 \( 1 + 128 T + p^{3} T^{2} \)
37 \( 1 + 26 T + p^{3} T^{2} \)
41 \( 1 + 10 T + p^{3} T^{2} \)
43 \( 1 - 52 T + p^{3} T^{2} \)
47 \( 1 - 544 T + p^{3} T^{2} \)
53 \( 1 + 6 p T + p^{3} T^{2} \)
59 \( 1 - 48 T + p^{3} T^{2} \)
61 \( 1 - 466 T + p^{3} T^{2} \)
67 \( 1 - 516 T + p^{3} T^{2} \)
71 \( 1 - 392 T + p^{3} T^{2} \)
73 \( 1 - 754 T + p^{3} T^{2} \)
79 \( 1 + p^{3} T^{2} \)
83 \( 1 + 624 T + p^{3} T^{2} \)
89 \( 1 - 1590 T + p^{3} T^{2} \)
97 \( 1 - 1018 T + p^{3} T^{2} \)
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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

−9.203894298850877146025815253694, −8.311673116357176531058742209062, −7.46932386035642504146176784253, −6.64328751215213016611982411037, −5.83364346769184833016832634986, −5.03197101279504367249844564287, −3.99337525618611805006101131399, −3.24467506311083167341188943444, −2.17971905077352842321075225407, −0.802096479533121950393797408254, 0.802096479533121950393797408254, 2.17971905077352842321075225407, 3.24467506311083167341188943444, 3.99337525618611805006101131399, 5.03197101279504367249844564287, 5.83364346769184833016832634986, 6.64328751215213016611982411037, 7.46932386035642504146176784253, 8.311673116357176531058742209062, 9.203894298850877146025815253694

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