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Properties

Label 2-1386-77.10-c1-0-22
Degree $2$
Conductor $1386$
Sign $0.856 + 0.515i$
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 + (0.194 − 0.112i)5-s + (2.62 + 0.307i)7-s − 0.999i·8-s + (0.112 − 0.194i)10-s + (3.31 + 0.0941i)11-s + 5.03·13-s + (2.42 − 1.04i)14-s + (−0.5 − 0.866i)16-s + (−3.92 + 6.80i)17-s + (−3.66 − 6.34i)19-s − 0.224i·20-s + (2.91 − 1.57i)22-s + (2.42 + 4.20i)23-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.249 − 0.433i)4-s + (0.0868 − 0.0501i)5-s + (0.993 + 0.116i)7-s − 0.353i·8-s + (0.0354 − 0.0613i)10-s + (0.999 + 0.0283i)11-s + 1.39·13-s + (0.649 − 0.280i)14-s + (−0.125 − 0.216i)16-s + (−0.952 + 1.65i)17-s + (−0.840 − 1.45i)19-s − 0.0501i·20-s + (0.622 − 0.336i)22-s + (0.506 + 0.877i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.856 + 0.515i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.856 + 0.515i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1386\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 11\)
Sign: $0.856 + 0.515i$
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: \((2,\ 1386,\ (\ :1/2),\ 0.856 + 0.515i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.949251652\)
\(L(\frac12)\) \(\approx\) \(2.949251652\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.62 - 0.307i)T \)
11 \( 1 + (-3.31 - 0.0941i)T \)
good5 \( 1 + (-0.194 + 0.112i)T + (2.5 - 4.33i)T^{2} \)
13 \( 1 - 5.03T + 13T^{2} \)
17 \( 1 + (3.92 - 6.80i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.66 + 6.34i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.42 - 4.20i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 5.65iT - 29T^{2} \)
31 \( 1 + (-3.73 - 2.15i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.0891 + 0.154i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 0.568T + 41T^{2} \)
43 \( 1 + 6.39iT - 43T^{2} \)
47 \( 1 + (2.93 - 1.69i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-1.07 + 1.86i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2.00 - 1.15i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.19 + 3.80i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.32 + 7.49i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 11.6T + 71T^{2} \)
73 \( 1 + (-5.25 + 9.10i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-8.75 + 5.05i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 14.2T + 83T^{2} \)
89 \( 1 + (9.10 - 5.25i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 3.96iT - 97T^{2} \)
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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

−9.399773975299672350012522715356, −8.747732028008801207569871029481, −8.045553530278352947044324424917, −6.74360219399593975166947494142, −6.21538490398558838874643206772, −5.21965478348804115389137500207, −4.24006110784654110052845163795, −3.67279364565377138068303840946, −2.15266766392007570520152433342, −1.31128773442280939603570482533, 1.31109143005511132586021501472, 2.56467878909787196401488073115, 3.91922617857970236036946538811, 4.45199037126392335897565854267, 5.48114846436349088856727650981, 6.42767665667019443616722020962, 6.93379406780049509308766843514, 8.202430597370177502974345589803, 8.521142669676487694384341339933, 9.534781134941130421688647018024

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