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Properties

Label 2-1386-77.54-c1-0-6
Degree $2$
Conductor $1386$
Sign $-0.794 - 0.607i$
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 + (−2.53 − 1.46i)5-s + (−2.00 − 1.72i)7-s + 0.999i·8-s + (−1.46 − 2.53i)10-s + (3.12 + 1.10i)11-s − 1.09·13-s + (−0.874 − 2.49i)14-s + (−0.5 + 0.866i)16-s + (2.00 + 3.47i)17-s + (−3.32 + 5.76i)19-s − 2.92i·20-s + (2.15 + 2.51i)22-s + (−0.874 + 1.51i)23-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.249 + 0.433i)4-s + (−1.13 − 0.653i)5-s + (−0.758 − 0.652i)7-s + 0.353i·8-s + (−0.461 − 0.800i)10-s + (0.943 + 0.332i)11-s − 0.302·13-s + (−0.233 − 0.667i)14-s + (−0.125 + 0.216i)16-s + (0.485 + 0.841i)17-s + (−0.763 + 1.32i)19-s − 0.653i·20-s + (0.459 + 0.537i)22-s + (−0.182 + 0.315i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.794 - 0.607i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.794 - 0.607i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1386\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 11\)
Sign: $-0.794 - 0.607i$
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: \((2,\ 1386,\ (\ :1/2),\ -0.794 - 0.607i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7740413754\)
\(L(\frac12)\) \(\approx\) \(0.7740413754\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.00 + 1.72i)T \)
11 \( 1 + (-3.12 - 1.10i)T \)
good5 \( 1 + (2.53 + 1.46i)T + (2.5 + 4.33i)T^{2} \)
13 \( 1 + 1.09T + 13T^{2} \)
17 \( 1 + (-2.00 - 3.47i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.32 - 5.76i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.874 - 1.51i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 6.40iT - 29T^{2} \)
31 \( 1 + (7.56 - 4.36i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (5.36 - 9.28i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 4.63T + 41T^{2} \)
43 \( 1 - 1.44iT - 43T^{2} \)
47 \( 1 + (-1.67 - 0.969i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-3.32 - 5.75i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (6.85 - 3.95i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.37 - 2.38i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.70 + 2.94i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 13.1T + 71T^{2} \)
73 \( 1 + (4.01 + 6.94i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (7.41 + 4.27i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 8.82T + 83T^{2} \)
89 \( 1 + (-6.94 - 4.01i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 12.9iT - 97T^{2} \)
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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

−9.933756506227701340770081855531, −8.907344271663368253178226771408, −8.098153098443027396066198476744, −7.46945027455561526134117556697, −6.59816106466427550253938877800, −5.83556271449499094078269208971, −4.59833347751835197481980223596, −3.93368850836182129171002955253, −3.40550369019524065079956838351, −1.54708580997195213794933355462, 0.25290512667046074631626222033, 2.25084600862638803478506068854, 3.28627598896047786131568895474, 3.81016338111222973781131499393, 4.94408905003626611038875549788, 5.90977005756037282443074243497, 6.95280798932100451064247498521, 7.21950038525888382347021259404, 8.646092520606679017165502436960, 9.216208197788007478344610541626

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