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Properties

Label 2-1386-77.10-c1-0-11
Degree $2$
Conductor $1386$
Sign $0.688 - 0.724i$
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 + (−1.92 + 1.11i)5-s + (2.45 + 0.982i)7-s − 0.999i·8-s + (−1.11 + 1.92i)10-s + (0.0571 + 3.31i)11-s − 0.112·13-s + (2.61 − 0.377i)14-s + (−0.5 − 0.866i)16-s + (−0.119 + 0.207i)17-s + (−0.218 − 0.377i)19-s + 2.22i·20-s + (1.70 + 2.84i)22-s + (0.401 + 0.695i)23-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.249 − 0.433i)4-s + (−0.861 + 0.497i)5-s + (0.928 + 0.371i)7-s − 0.353i·8-s + (−0.351 + 0.609i)10-s + (0.0172 + 0.999i)11-s − 0.0312·13-s + (0.699 − 0.100i)14-s + (−0.125 − 0.216i)16-s + (−0.0290 + 0.0503i)17-s + (−0.0500 − 0.0866i)19-s + 0.497i·20-s + (0.364 + 0.606i)22-s + (0.0837 + 0.145i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.688 - 0.724i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.688 - 0.724i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1386\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 11\)
Sign: $0.688 - 0.724i$
Analytic conductor: \(11.0672\)
Root analytic conductor: \(3.32675\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1386} (703, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1386,\ (\ :1/2),\ 0.688 - 0.724i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.048488081\)
\(L(\frac12)\) \(\approx\) \(2.048488081\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.45 - 0.982i)T \)
11 \( 1 + (-0.0571 - 3.31i)T \)
good5 \( 1 + (1.92 - 1.11i)T + (2.5 - 4.33i)T^{2} \)
13 \( 1 + 0.112T + 13T^{2} \)
17 \( 1 + (0.119 - 0.207i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.218 + 0.377i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.401 - 0.695i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 7.50iT - 29T^{2} \)
31 \( 1 + (-0.306 - 0.177i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-3.67 - 6.36i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 3.14T + 41T^{2} \)
43 \( 1 - 10.1iT - 43T^{2} \)
47 \( 1 + (-11.2 + 6.47i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (1.28 - 2.22i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (3.27 + 1.89i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.524 - 0.909i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.48 + 4.29i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 7.58T + 71T^{2} \)
73 \( 1 + (-2.39 + 4.15i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.429 + 0.247i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 - 0.569T + 83T^{2} \)
89 \( 1 + (-12.1 + 6.99i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 10.6iT - 97T^{2} \)
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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

−9.815721930627220086398652413458, −8.876464422834888777766141081736, −7.894087248759461878297563563087, −7.28127571615949217071143464071, −6.40839882709986699543836690084, −5.21346167346150453630201734570, −4.60153301431911388611112457400, −3.67262011159094319564830300080, −2.63757445112143814321972203455, −1.50323674596426064937880887605, 0.71880805460705748272143546061, 2.37654015043317680231738813248, 3.77975491997755784647727016988, 4.26419856158150610208816704885, 5.23831914967135447890361457936, 6.02028019429081468461156839447, 7.12868350181895771529999618825, 7.930937031018604004554626448708, 8.335327086358041210566356112709, 9.230974334353277121592523522682

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