Properties

Label 2-1386-21.20-c1-0-27
Degree $2$
Conductor $1386$
Sign $-0.0523 + 0.998i$
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  + 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

\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0523 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0523 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1386\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 11\)
Sign: $-0.0523 + 0.998i$
Analytic conductor: \(11.0672\)
Root analytic conductor: \(3.32675\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1386} (881, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1386,\ (\ :1/2),\ -0.0523 + 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6149948469\)
\(L(\frac12)\) \(\approx\) \(0.6149948469\)
\(L(\frac{3}{2})\) 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

−9.616692430225485365789504549196, −8.310216966442115966974285251321, −7.85320340598134582747319891128, −6.69498838651167334543651160987, −6.23932144138914177020585190134, −5.40687299885042985088687147085, −4.19992945404993743800534825904, −3.49766467689644574256715391732, −2.00412784936308371621538383241, −0.23962020225016066391790956021, 1.60222225534461171303534266997, 2.69670092259974169606776059556, 3.47118956810352184933443199300, 4.76350709135407470611765993586, 5.45388163549011253809594005566, 6.55061747485704364528393954632, 7.20594376333783704073141459509, 8.621507490930312411826907581309, 9.040050939115512683272694284726, 9.790202759411511018132792599972

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