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Properties

Label 2-1386-77.54-c1-0-11
Degree $2$
Conductor $1386$
Sign $0.514 - 0.857i$
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 + (2.13 + 1.23i)5-s + (−0.941 − 2.47i)7-s − 0.999i·8-s + (−1.23 − 2.13i)10-s + (0.881 + 3.19i)11-s − 1.32·13-s + (−0.420 + 2.61i)14-s + (−0.5 + 0.866i)16-s + (2.23 + 3.87i)17-s + (−2.21 + 3.83i)19-s + 2.46i·20-s + (0.834 − 3.20i)22-s + (−4.14 + 7.17i)23-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (0.249 + 0.433i)4-s + (0.953 + 0.550i)5-s + (−0.356 − 0.934i)7-s − 0.353i·8-s + (−0.389 − 0.674i)10-s + (0.265 + 0.964i)11-s − 0.366·13-s + (−0.112 + 0.698i)14-s + (−0.125 + 0.216i)16-s + (0.542 + 0.939i)17-s + (−0.508 + 0.880i)19-s + 0.550i·20-s + (0.178 − 0.684i)22-s + (−0.864 + 1.49i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.514 - 0.857i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.514 - 0.857i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1386\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 11\)
Sign: $0.514 - 0.857i$
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: \((2,\ 1386,\ (\ :1/2),\ 0.514 - 0.857i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.200348007\)
\(L(\frac12)\) \(\approx\) \(1.200348007\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.941 + 2.47i)T \)
11 \( 1 + (-0.881 - 3.19i)T \)
good5 \( 1 + (-2.13 - 1.23i)T + (2.5 + 4.33i)T^{2} \)
13 \( 1 + 1.32T + 13T^{2} \)
17 \( 1 + (-2.23 - 3.87i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.21 - 3.83i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (4.14 - 7.17i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 1.44iT - 29T^{2} \)
31 \( 1 + (-2.34 + 1.35i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.46 + 4.27i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 8.18T + 41T^{2} \)
43 \( 1 - 9.63iT - 43T^{2} \)
47 \( 1 + (-0.664 - 0.383i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.945 + 1.63i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (6.74 - 3.89i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.23 - 3.87i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.861 - 1.49i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 10.3T + 71T^{2} \)
73 \( 1 + (5.58 + 9.67i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-2.53 - 1.46i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 10.1T + 83T^{2} \)
89 \( 1 + (-10.9 - 6.32i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 1.37iT - 97T^{2} \)
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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

−9.848722814381976745128714888139, −9.292121293265163501541877980760, −7.86751318300904966141843095556, −7.53037320571832165485668412975, −6.40272608568100354302574827585, −5.90616225812692175421609216694, −4.37735079908288965602684165121, −3.56729766142245825167143539349, −2.30606594932842398585716599884, −1.41734624057615607239261508702, 0.60684753255233630696671971933, 2.10234721037960028494415191357, 2.96610388011286497108772476231, 4.63823594261387196989525974414, 5.52314274731278059038190315172, 6.12924289999174191278231432324, 6.87280310378009693655437942796, 8.086124398420116447052956181783, 8.794064117612907565774124350488, 9.293866542688363928074458640613

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