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Properties

Label 2-1386-77.10-c1-0-8
Degree $2$
Conductor $1386$
Sign $0.751 - 0.660i$
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 + (−2.83 + 1.63i)5-s + (−2.54 − 0.732i)7-s − 0.999i·8-s + (−1.63 + 2.83i)10-s + (0.569 − 3.26i)11-s + 5.12·13-s + (−2.56 + 0.637i)14-s + (−0.5 − 0.866i)16-s + (−2.66 + 4.61i)17-s + (2.13 + 3.70i)19-s + 3.26i·20-s + (−1.14 − 3.11i)22-s + (2.68 + 4.65i)23-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.249 − 0.433i)4-s + (−1.26 + 0.730i)5-s + (−0.960 − 0.276i)7-s − 0.353i·8-s + (−0.516 + 0.895i)10-s + (0.171 − 0.985i)11-s + 1.42·13-s + (−0.686 + 0.170i)14-s + (−0.125 − 0.216i)16-s + (−0.645 + 1.11i)17-s + (0.490 + 0.849i)19-s + 0.730i·20-s + (−0.243 − 0.663i)22-s + (0.560 + 0.970i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.751 - 0.660i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.751 - 0.660i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1386\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 11\)
Sign: $0.751 - 0.660i$
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: \((2,\ 1386,\ (\ :1/2),\ 0.751 - 0.660i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.496325471\)
\(L(\frac12)\) \(\approx\) \(1.496325471\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.54 + 0.732i)T \)
11 \( 1 + (-0.569 + 3.26i)T \)
good5 \( 1 + (2.83 - 1.63i)T + (2.5 - 4.33i)T^{2} \)
13 \( 1 - 5.12T + 13T^{2} \)
17 \( 1 + (2.66 - 4.61i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.13 - 3.70i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.68 - 4.65i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 4.77iT - 29T^{2} \)
31 \( 1 + (-5.59 - 3.23i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.06 + 1.84i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 0.949T + 41T^{2} \)
43 \( 1 - 6.20iT - 43T^{2} \)
47 \( 1 + (10.4 - 6.01i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-6.26 + 10.8i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-11.6 - 6.74i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.67 + 4.62i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.09 - 5.36i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 10.8T + 71T^{2} \)
73 \( 1 + (6.61 - 11.4i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (7.93 - 4.58i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 - 0.835T + 83T^{2} \)
89 \( 1 + (-5.39 + 3.11i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 0.624iT - 97T^{2} \)
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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

−9.921517852950067782567501096963, −8.724072869367984767867113692927, −8.105046955081483795931879812708, −6.98976905012731326660666160968, −6.40457368673887070898136776723, −5.62064602978631185925331195550, −4.13067042872815831520838116789, −3.51868530202873482127566204459, −3.11452115076591112255375753303, −1.19996374800204335085928067247, 0.57740557608306727496036244523, 2.56812010656090633142592199171, 3.62479720364888590716301134906, 4.40129092053573375587663954541, 5.07864011119480028513318045364, 6.33453373728121501674042196510, 6.92428232001482157042930023002, 7.77134349784004167912554039284, 8.694719066026317249686021633782, 9.179810579103039030401266585546

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