Properties

Degree $2$
Conductor $1386$
Sign $0.213 + 0.976i$
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 + 2.01·5-s + (1.78 − 1.95i)7-s + i·8-s − 2.01i·10-s + i·11-s + (−1.95 − 1.78i)14-s + 16-s + 5.92·17-s + 0.657i·19-s − 2.01·20-s + 22-s − 7.87i·23-s − 0.935·25-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.5·4-s + 0.901·5-s + (0.674 − 0.738i)7-s + 0.353i·8-s − 0.637i·10-s + 0.301i·11-s + (−0.522 − 0.476i)14-s + 0.250·16-s + 1.43·17-s + 0.150i·19-s − 0.450·20-s + 0.213·22-s − 1.64i·23-s − 0.187·25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.213 + 0.976i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.213 + 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1386\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 11\)
Sign: $0.213 + 0.976i$
Motivic weight: \(1\)
Character: $\chi_{1386} (881, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1386,\ (\ :1/2),\ 0.213 + 0.976i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.086018432\)
\(L(\frac12)\) \(\approx\) \(2.086018432\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.78 + 1.95i)T \)
11 \( 1 - iT \)
good5 \( 1 - 2.01T + 5T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 - 5.92T + 17T^{2} \)
19 \( 1 - 0.657iT - 19T^{2} \)
23 \( 1 + 7.87iT - 23T^{2} \)
29 \( 1 + 1.32iT - 29T^{2} \)
31 \( 1 - 10.6iT - 31T^{2} \)
37 \( 1 - 5.87T + 37T^{2} \)
41 \( 1 - 4.56T + 41T^{2} \)
43 \( 1 + 1.56T + 43T^{2} \)
47 \( 1 - 0.532T + 47T^{2} \)
53 \( 1 + 7.44iT - 53T^{2} \)
59 \( 1 - 4.03T + 59T^{2} \)
61 \( 1 - 0.250iT - 61T^{2} \)
67 \( 1 + 7.81T + 67T^{2} \)
71 \( 1 - 0.609iT - 71T^{2} \)
73 \( 1 - 0.532iT - 73T^{2} \)
79 \( 1 + 12.7T + 79T^{2} \)
83 \( 1 - 13.3T + 83T^{2} \)
89 \( 1 - 5.22T + 89T^{2} \)
97 \( 1 + 2.71iT - 97T^{2} \)
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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

−9.702228353093015400932007284894, −8.670626935592757783750502832031, −7.940112172960588469658862979738, −6.99473303409172527394417651159, −5.95479301861980291989321626238, −5.06435621358543080731623232745, −4.27062555326385726278166215689, −3.13976176847647145786769911347, −2.01019586586484744710829241741, −1.01976969820193861913893217472, 1.33006595297265707933620384409, 2.56112098894823565802724792774, 3.83616725137395097998160707075, 5.06134140911464057333975809389, 5.75966292830878564142809831419, 6.12528931148703414221772533010, 7.60478467019353465985745996021, 7.86403665839867295833346695349, 9.074150689739232067140494342814, 9.490287634749236902580162175001

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