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Properties

Label 2-1386-21.20-c1-0-6
Degree $2$
Conductor $1386$
Sign $-0.933 - 0.359i$
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 + 2.79·5-s + (−2.20 + 1.46i)7-s − i·8-s + 2.79i·10-s + i·11-s + 4.14i·13-s + (−1.46 − 2.20i)14-s + 16-s − 5.73·17-s − 1.15i·19-s − 2.79·20-s − 22-s + 2.83·25-s − 4.14·26-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.5·4-s + 1.25·5-s + (−0.832 + 0.554i)7-s − 0.353i·8-s + 0.885i·10-s + 0.301i·11-s + 1.15i·13-s + (−0.391 − 0.588i)14-s + 0.250·16-s − 1.39·17-s − 0.266i·19-s − 0.626·20-s − 0.213·22-s + 0.567·25-s − 0.813·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.933 - 0.359i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.933 - 0.359i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1386\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 11\)
Sign: $-0.933 - 0.359i$
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: \((2,\ 1386,\ (\ :1/2),\ -0.933 - 0.359i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.237071116\)
\(L(\frac12)\) \(\approx\) \(1.237071116\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.20 - 1.46i)T \)
11 \( 1 - iT \)
good5 \( 1 - 2.79T + 5T^{2} \)
13 \( 1 - 4.14iT - 13T^{2} \)
17 \( 1 + 5.73T + 17T^{2} \)
19 \( 1 + 1.15iT - 19T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 - 10.0iT - 29T^{2} \)
31 \( 1 + 1.77iT - 31T^{2} \)
37 \( 1 - 1.40T + 37T^{2} \)
41 \( 1 + 2.37T + 41T^{2} \)
43 \( 1 + 7.00T + 43T^{2} \)
47 \( 1 - 9.69T + 47T^{2} \)
53 \( 1 - 11.8iT - 53T^{2} \)
59 \( 1 + 13.8T + 59T^{2} \)
61 \( 1 + 1.71iT - 61T^{2} \)
67 \( 1 - 12.0T + 67T^{2} \)
71 \( 1 - 8.48iT - 71T^{2} \)
73 \( 1 - 10.2iT - 73T^{2} \)
79 \( 1 - 1.43T + 79T^{2} \)
83 \( 1 + 7.55T + 83T^{2} \)
89 \( 1 + 2.33T + 89T^{2} \)
97 \( 1 - 7.91iT - 97T^{2} \)
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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

−9.606780434446764268430712828469, −9.125587154107399799018601679474, −8.612200147863814733688431967579, −7.12007173458702048752785926648, −6.65400293986374154941343503004, −5.95694998625395374632732256938, −5.12039187957366452522071144611, −4.16137660521931651559853530095, −2.76763514040944971789735551535, −1.75571355858812429980630314109, 0.47374750025638383676536655349, 1.96462375109688616397068720914, 2.87435980477866400167894983815, 3.86702618632861344358411854278, 4.97243083126414262812745800331, 5.97147720492743263615251718153, 6.49574780685325880764414363365, 7.70317687203460716767545301517, 8.665427765378161453337801383803, 9.491882975165225179842656248021

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