Properties

Degree $2$
Conductor $1386$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s + 3.80·5-s + 7-s + 8-s + 3.80·10-s + 11-s − 6.44·13-s + 14-s + 16-s + 3.80·17-s + 6.64·19-s + 3.80·20-s + 22-s + 0.842·23-s + 9.44·25-s − 6.44·26-s + 28-s − 8.44·29-s − 9.40·31-s + 32-s + 3.80·34-s + 3.80·35-s + 9.60·37-s + 6.64·38-s + 3.80·40-s − 3.80·41-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s + 1.69·5-s + 0.377·7-s + 0.353·8-s + 1.20·10-s + 0.301·11-s − 1.78·13-s + 0.267·14-s + 0.250·16-s + 0.921·17-s + 1.52·19-s + 0.849·20-s + 0.213·22-s + 0.175·23-s + 1.88·25-s − 1.26·26-s + 0.188·28-s − 1.56·29-s − 1.68·31-s + 0.176·32-s + 0.651·34-s + 0.642·35-s + 1.57·37-s + 1.07·38-s + 0.600·40-s − 0.593·41-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: $1$
Motivic weight: \(1\)
Character: $\chi_{1386} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1386,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.615541701\)
\(L(\frac12)\) \(\approx\) \(3.615541701\)
\(L(\frac{3}{2})\) 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 \( 1 - 3.80T + 5T^{2} \)
13 \( 1 + 6.44T + 13T^{2} \)
17 \( 1 - 3.80T + 17T^{2} \)
19 \( 1 - 6.64T + 19T^{2} \)
23 \( 1 - 0.842T + 23T^{2} \)
29 \( 1 + 8.44T + 29T^{2} \)
31 \( 1 + 9.40T + 31T^{2} \)
37 \( 1 - 9.60T + 37T^{2} \)
41 \( 1 + 3.80T + 41T^{2} \)
43 \( 1 + 8.04T + 43T^{2} \)
47 \( 1 - 6.24T + 47T^{2} \)
53 \( 1 + 1.60T + 53T^{2} \)
59 \( 1 + 9.28T + 59T^{2} \)
61 \( 1 - 8.75T + 61T^{2} \)
67 \( 1 - 1.15T + 67T^{2} \)
71 \( 1 + 6.75T + 71T^{2} \)
73 \( 1 + 7.80T + 73T^{2} \)
79 \( 1 - 2.84T + 79T^{2} \)
83 \( 1 + 9.80T + 83T^{2} \)
89 \( 1 + 5.15T + 89T^{2} \)
97 \( 1 - 18.8T + 97T^{2} \)
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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

−9.701522198182578327904401679827, −9.086431194279069426066051570268, −7.57757689610496881944711607465, −7.17395496757360296755173465586, −5.93753404362871426181355861379, −5.42194327190271186074624051041, −4.81467675683958722081657223521, −3.38001615177652481692593974125, −2.37487282676375360395858562462, −1.49587078039262481950061262573, 1.49587078039262481950061262573, 2.37487282676375360395858562462, 3.38001615177652481692593974125, 4.81467675683958722081657223521, 5.42194327190271186074624051041, 5.93753404362871426181355861379, 7.17395496757360296755173465586, 7.57757689610496881944711607465, 9.086431194279069426066051570268, 9.701522198182578327904401679827

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