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Properties

Label 2-1386-33.29-c1-0-11
Degree $2$
Conductor $1386$
Sign $-0.382 - 0.923i$
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  + (0.809 + 0.587i)2-s + (0.309 + 0.951i)4-s + (2.22 + 3.06i)5-s + (0.951 − 0.309i)7-s + (−0.309 + 0.951i)8-s + 3.79i·10-s + (−1.49 + 2.96i)11-s + (1.70 − 2.35i)13-s + (0.951 + 0.309i)14-s + (−0.809 + 0.587i)16-s + (−5.61 + 4.08i)17-s + (7.19 + 2.33i)19-s + (−2.22 + 3.06i)20-s + (−2.94 + 1.51i)22-s − 7.90i·23-s + ⋯
L(s)  = 1  + (0.572 + 0.415i)2-s + (0.154 + 0.475i)4-s + (0.996 + 1.37i)5-s + (0.359 − 0.116i)7-s + (−0.109 + 0.336i)8-s + 1.19i·10-s + (−0.450 + 0.892i)11-s + (0.474 − 0.652i)13-s + (0.254 + 0.0825i)14-s + (−0.202 + 0.146i)16-s + (−1.36 + 0.989i)17-s + (1.65 + 0.536i)19-s + (−0.498 + 0.685i)20-s + (−0.628 + 0.323i)22-s − 1.64i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.382 - 0.923i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.382 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1386\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 11\)
Sign: $-0.382 - 0.923i$
Analytic conductor: \(11.0672\)
Root analytic conductor: \(3.32675\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1386} (953, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1386,\ (\ :1/2),\ -0.382 - 0.923i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.752388474\)
\(L(\frac12)\) \(\approx\) \(2.752388474\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.809 - 0.587i)T \)
3 \( 1 \)
7 \( 1 + (-0.951 + 0.309i)T \)
11 \( 1 + (1.49 - 2.96i)T \)
good5 \( 1 + (-2.22 - 3.06i)T + (-1.54 + 4.75i)T^{2} \)
13 \( 1 + (-1.70 + 2.35i)T + (-4.01 - 12.3i)T^{2} \)
17 \( 1 + (5.61 - 4.08i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (-7.19 - 2.33i)T + (15.3 + 11.1i)T^{2} \)
23 \( 1 + 7.90iT - 23T^{2} \)
29 \( 1 + (-0.983 - 3.02i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (2.88 + 2.09i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (1.62 + 4.98i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (0.236 - 0.728i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 1.48iT - 43T^{2} \)
47 \( 1 + (-7.03 - 2.28i)T + (38.0 + 27.6i)T^{2} \)
53 \( 1 + (2.22 - 3.05i)T + (-16.3 - 50.4i)T^{2} \)
59 \( 1 + (4.46 - 1.44i)T + (47.7 - 34.6i)T^{2} \)
61 \( 1 + (7.07 + 9.73i)T + (-18.8 + 58.0i)T^{2} \)
67 \( 1 - 6.51T + 67T^{2} \)
71 \( 1 + (-0.922 - 1.27i)T + (-21.9 + 67.5i)T^{2} \)
73 \( 1 + (-9.04 + 2.94i)T + (59.0 - 42.9i)T^{2} \)
79 \( 1 + (6.86 - 9.44i)T + (-24.4 - 75.1i)T^{2} \)
83 \( 1 + (-10.7 + 7.83i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 - 1.72iT - 89T^{2} \)
97 \( 1 + (-0.0144 - 0.0105i)T + (29.9 + 92.2i)T^{2} \)
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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

−9.985166040562118244682628657614, −9.025706029161395734668801822401, −7.948195949468714165687525881421, −7.21764549679654968617102472930, −6.48089874414257513971013638707, −5.81812456010779418773676356805, −4.92541492314618580229125099359, −3.79630716846834758625390508723, −2.75487829533455559131532090164, −1.92223353087440669720639682953, 0.951628566111689551784612325092, 1.93721966048738931365918403556, 3.10644587098692929346687087539, 4.37940115157688631214937741463, 5.22847334155194803423792394192, 5.57244613625706745977245961237, 6.64937823011131215037315802393, 7.77182277233060139835186382589, 8.946811745072462495819734769532, 9.173258865614629255256210178736

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