Properties

Degree $2$
Conductor $1386$
Sign $0.958 + 0.283i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + i·2-s − 4-s − 0.929·5-s + (−2.07 + 1.63i)7-s i·8-s − 0.929i·10-s i·11-s + (−1.63 − 2.07i)14-s + 16-s + 2.34·17-s − 6.87i·19-s + 0.929·20-s + 22-s − 3.04i·23-s − 4.13·25-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.5·4-s − 0.415·5-s + (−0.785 + 0.619i)7-s − 0.353i·8-s − 0.294i·10-s − 0.301i·11-s + (−0.437 − 0.555i)14-s + 0.250·16-s + 0.569·17-s − 1.57i·19-s + 0.207·20-s + 0.213·22-s − 0.635i·23-s − 0.827·25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.958 + 0.283i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.958 + 0.283i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1386\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 11\)
Sign: $0.958 + 0.283i$
Motivic weight: \(1\)
Character: $\chi_{1386} (881, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1386,\ (\ :1/2),\ 0.958 + 0.283i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9911541602\)
\(L(\frac12)\) \(\approx\) \(0.9911541602\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
3 \( 1 \)
7 \( 1 + (2.07 - 1.63i)T \)
11 \( 1 + iT \)
good5 \( 1 + 0.929T + 5T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 - 2.34T + 17T^{2} \)
19 \( 1 + 6.87iT - 19T^{2} \)
23 \( 1 + 3.04iT - 23T^{2} \)
29 \( 1 + 6.39iT - 29T^{2} \)
31 \( 1 - 10.0iT - 31T^{2} \)
37 \( 1 + 5.04T + 37T^{2} \)
41 \( 1 - 10.1T + 41T^{2} \)
43 \( 1 - 6.15T + 43T^{2} \)
47 \( 1 - 12.0T + 47T^{2} \)
53 \( 1 + 11.2iT - 53T^{2} \)
59 \( 1 + 1.85T + 59T^{2} \)
61 \( 1 - 10.2iT - 61T^{2} \)
67 \( 1 + 0.0878T + 67T^{2} \)
71 \( 1 + 11.5iT - 71T^{2} \)
73 \( 1 + 12.0iT - 73T^{2} \)
79 \( 1 - 13.5T + 79T^{2} \)
83 \( 1 + 4.06T + 83T^{2} \)
89 \( 1 - 17.0T + 89T^{2} \)
97 \( 1 + 15.6iT - 97T^{2} \)
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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

−9.223868130625018907493229407161, −8.821151083270053985002468586618, −7.84689976361186353156193807878, −7.07724719040026496714932520636, −6.27358921555073563231201770339, −5.53157706482099015139187457267, −4.55567921415913462738454157021, −3.52204238362193075613537981422, −2.51482080409563649379995777178, −0.48279924590825292288324225344, 1.09413389785414146625048930157, 2.48072636785076359739177355217, 3.77256139920138794970030946774, 3.98706438061649028334489964502, 5.43998830676597376055175431950, 6.20481543679167642135611798286, 7.52079789816293653756120192809, 7.79305893743532675439174257360, 9.116107482911826794140995892761, 9.674213966643654799726034791029

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