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Properties

Label 2-1386-7.2-c1-0-19
Degree $2$
Conductor $1386$
Sign $0.895 + 0.444i$
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  + (−0.5 + 0.866i)2-s + (−0.499 − 0.866i)4-s + (−1.32 + 2.29i)5-s + (1.32 − 2.29i)7-s + 0.999·8-s + (−1.32 − 2.29i)10-s + (0.5 + 0.866i)11-s − 4·13-s + (1.32 + 2.29i)14-s + (−0.5 + 0.866i)16-s + (−1.5 − 2.59i)17-s + (2.64 − 4.58i)19-s + 2.64·20-s − 0.999·22-s + (1.32 − 2.29i)23-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + (−0.591 + 1.02i)5-s + (0.499 − 0.866i)7-s + 0.353·8-s + (−0.418 − 0.724i)10-s + (0.150 + 0.261i)11-s − 1.10·13-s + (0.353 + 0.612i)14-s + (−0.125 + 0.216i)16-s + (−0.363 − 0.630i)17-s + (0.606 − 1.05i)19-s + 0.591·20-s − 0.213·22-s + (0.275 − 0.477i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.895 + 0.444i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.895 + 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1386\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 11\)
Sign: $0.895 + 0.444i$
Analytic conductor: \(11.0672\)
Root analytic conductor: \(3.32675\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1386} (793, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1386,\ (\ :1/2),\ 0.895 + 0.444i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9313844405\)
\(L(\frac12)\) \(\approx\) \(0.9313844405\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 - 0.866i)T \)
3 \( 1 \)
7 \( 1 + (-1.32 + 2.29i)T \)
11 \( 1 + (-0.5 - 0.866i)T \)
good5 \( 1 + (1.32 - 2.29i)T + (-2.5 - 4.33i)T^{2} \)
13 \( 1 + 4T + 13T^{2} \)
17 \( 1 + (1.5 + 2.59i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.64 + 4.58i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.32 + 2.29i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 2T + 29T^{2} \)
31 \( 1 + (-2 - 3.46i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-4.64 + 8.04i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 9T + 41T^{2} \)
43 \( 1 + 1.29T + 43T^{2} \)
47 \( 1 + (-5.96 + 10.3i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-2 - 3.46i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (7.29 + 12.6i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.96 + 3.40i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6.79 - 11.7i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 13.2T + 71T^{2} \)
73 \( 1 + (-2.35 - 4.07i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (2.67 - 4.63i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 5.58T + 83T^{2} \)
89 \( 1 + (-6.64 + 11.5i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 9.58T + 97T^{2} \)
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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

−9.557372328468437466206838075877, −8.584369337253576649970554850730, −7.61466618544954241172188084215, −7.06975611597040934509829409975, −6.77146619282409019297659817981, −5.24990828506805748583647289511, −4.57556478392073167052375579857, −3.47804850656669731582976876660, −2.28908451475171467139040390514, −0.48759280464895451795573031412, 1.16988108418741682822660645638, 2.29641256911200412026960426896, 3.51374270017800957239197826422, 4.57293072408368928818524285849, 5.19764663948056573929526434745, 6.26485777466146085967998960653, 7.72121738372156278983306331783, 8.056910677164682261252720135832, 8.919446841736475715357567856631, 9.490396069791913994735235965287

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