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Properties

Label 2-1386-77.10-c1-0-6
Degree $2$
Conductor $1386$
Sign $-0.0387 - 0.999i$
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 + (−1.73 − 2.82i)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 + (−0.524 − 0.851i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0387 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0387 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1386\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 11\)
Sign: $-0.0387 - 0.999i$
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: \((2,\ 1386,\ (\ :1/2),\ -0.0387 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6988624278\)
\(L(\frac12)\) \(\approx\) \(0.6988624278\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.73 + 2.82i)T \)
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

−9.717422952800168036809554755187, −9.219030347135094219411463097364, −7.985018363535810099460195792165, −7.44826869451263044279720746414, −6.72304381397950727734629911167, −5.52140540783014005157720599876, −5.20175545184663284973249513802, −3.44781418828540901515383140440, −2.77847291111234072236869024152, −1.06915805861394365701469786756, 0.40263023310572966412893628367, 2.20305479816057559778352169684, 2.88924258959849349386232831815, 4.11678509997300347137238276767, 5.14874307715399883762756670651, 6.26162326182242853379167496654, 7.06278718758006544670250682689, 7.77839409048247856826720582284, 8.646005000981597330389350528978, 9.728389141945830282874067498461

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