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Properties

Label 2-1386-77.54-c1-0-35
Degree $2$
Conductor $1386$
Sign $0.935 - 0.353i$
Analytic cond. $11.0672$
Root an. cond. $3.32675$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.866 + 0.5i)2-s + (0.499 + 0.866i)4-s + (3.06 + 1.76i)5-s + (−0.649 − 2.56i)7-s + 0.999i·8-s + (1.76 + 3.06i)10-s + (0.926 − 3.18i)11-s + 5.01·13-s + (0.720 − 2.54i)14-s + (−0.5 + 0.866i)16-s + (−1.94 − 3.37i)17-s + (2.32 − 4.03i)19-s + 3.53i·20-s + (2.39 − 2.29i)22-s + (−0.779 + 1.35i)23-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.249 + 0.433i)4-s + (1.37 + 0.791i)5-s + (−0.245 − 0.969i)7-s + 0.353i·8-s + (0.559 + 0.969i)10-s + (0.279 − 0.960i)11-s + 1.39·13-s + (0.192 − 0.680i)14-s + (−0.125 + 0.216i)16-s + (−0.472 − 0.817i)17-s + (0.534 − 0.925i)19-s + 0.791i·20-s + (0.510 − 0.489i)22-s + (−0.162 + 0.281i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.935 - 0.353i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.935 - 0.353i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1386\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 11\)
Sign: $0.935 - 0.353i$
Analytic conductor: \(11.0672\)
Root analytic conductor: \(3.32675\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1386} (901, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1386,\ (\ :1/2),\ 0.935 - 0.353i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.173570456\)
\(L(\frac12)\) \(\approx\) \(3.173570456\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 - 0.5i)T \)
3 \( 1 \)
7 \( 1 + (0.649 + 2.56i)T \)
11 \( 1 + (-0.926 + 3.18i)T \)
good5 \( 1 + (-3.06 - 1.76i)T + (2.5 + 4.33i)T^{2} \)
13 \( 1 - 5.01T + 13T^{2} \)
17 \( 1 + (1.94 + 3.37i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.32 + 4.03i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.779 - 1.35i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 0.100iT - 29T^{2} \)
31 \( 1 + (0.242 - 0.139i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.352 - 0.610i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 2.94T + 41T^{2} \)
43 \( 1 - 9.03iT - 43T^{2} \)
47 \( 1 + (5.68 + 3.28i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-2.77 - 4.79i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (12.5 - 7.26i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.56 - 9.63i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.99 + 3.45i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 8.45T + 71T^{2} \)
73 \( 1 + (1.94 + 3.37i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (6.66 + 3.84i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 2.87T + 83T^{2} \)
89 \( 1 + (-5.15 - 2.97i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 16.7iT - 97T^{2} \)
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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

−9.545917660633375052384567310200, −8.959346221634043805268934769320, −7.77776475257801947157262156852, −6.86709999528732325277914202803, −6.30496450224702319154294790140, −5.71882670245620938768637820566, −4.58300197798286166466115719964, −3.44637497763601412861465252051, −2.76853252837691754518529063313, −1.25730134349072233124416155086, 1.53933516622361299939568175544, 2.07338425583083990484798244661, 3.43903690693754057642739880750, 4.51501590948596489761723516422, 5.46309958885486460195805296061, 6.01051862874169960982114849962, 6.62972884875862353210801489844, 8.150295557459756525096626769114, 8.940008030678323739979313969234, 9.551280678985701108402495192976

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