Properties

Label 2-1386-21.5-c1-0-3
Degree $2$
Conductor $1386$
Sign $0.410 - 0.912i$
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 + (1.22 − 2.12i)5-s + (−1 + 2.44i)7-s − 0.999i·8-s + (−2.12 + 1.22i)10-s + (−0.866 + 0.5i)11-s + 2.44i·13-s + (2.09 − 1.62i)14-s + (−0.5 + 0.866i)16-s + (0.866 + 1.5i)17-s + (−1.5 − 0.866i)19-s + 2.44·20-s + 0.999·22-s + (−1.07 − 0.621i)23-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (0.249 + 0.433i)4-s + (0.547 − 0.948i)5-s + (−0.377 + 0.925i)7-s − 0.353i·8-s + (−0.670 + 0.387i)10-s + (−0.261 + 0.150i)11-s + 0.679i·13-s + (0.558 − 0.433i)14-s + (−0.125 + 0.216i)16-s + (0.210 + 0.363i)17-s + (−0.344 − 0.198i)19-s + 0.547·20-s + 0.213·22-s + (−0.224 − 0.129i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.410 - 0.912i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.410 - 0.912i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1386\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 11\)
Sign: $0.410 - 0.912i$
Analytic conductor: \(11.0672\)
Root analytic conductor: \(3.32675\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1386} (89, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1386,\ (\ :1/2),\ 0.410 - 0.912i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9089232774\)
\(L(\frac12)\) \(\approx\) \(0.9089232774\)
\(L(\frac{3}{2})\) 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 \( 1 + (1 - 2.44i)T \)
11 \( 1 + (0.866 - 0.5i)T \)
good5 \( 1 + (-1.22 + 2.12i)T + (-2.5 - 4.33i)T^{2} \)
13 \( 1 - 2.44iT - 13T^{2} \)
17 \( 1 + (-0.866 - 1.5i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.5 + 0.866i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.07 + 0.621i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 7.24iT - 29T^{2} \)
31 \( 1 + (7.24 - 4.18i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.62 - 4.54i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 8.36T + 41T^{2} \)
43 \( 1 - T + 43T^{2} \)
47 \( 1 + (1.37 - 2.37i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (2.30 + 3.98i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-9 - 5.19i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.24 + 3.88i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 1.24iT - 71T^{2} \)
73 \( 1 + (4.24 - 2.44i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (4 - 6.92i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 7.94T + 83T^{2} \)
89 \( 1 + (-2.95 + 5.12i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 6.63iT - 97T^{2} \)
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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

−9.526024951445909016634482736809, −8.930501448848246347442066146070, −8.548476477379778955376012152234, −7.40270039900679333102873621304, −6.45674044701306664277197051271, −5.54988868575649781454133669258, −4.76230908977145404416609053270, −3.48574367099021206298892565246, −2.30680172020370092263422997709, −1.39137989403283734565056790995, 0.45492873854361321779354912078, 2.11169663435721733875371032799, 3.15028136939763680781396950707, 4.24288101465527373997218789787, 5.66768720051519828286214799695, 6.16800769161298754043182702861, 7.21094747845942552182034397946, 7.57021684937986702427255611667, 8.585408910169913929301937714097, 9.708484705143657379454642323884

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