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Properties

Label 2-1386-231.32-c1-0-26
Degree $2$
Conductor $1386$
Sign $0.999 - 0.0153i$
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  + (0.5 + 0.866i)2-s + (−0.499 + 0.866i)4-s + (1.78 − 1.02i)5-s + (0.289 + 2.62i)7-s − 0.999·8-s + (1.78 + 1.02i)10-s + (−1.29 − 3.05i)11-s − 6.97i·13-s + (−2.13 + 1.56i)14-s + (−0.5 − 0.866i)16-s + (2.66 − 4.61i)17-s + (7.36 − 4.25i)19-s + 2.05i·20-s + (1.99 − 2.64i)22-s + (1.35 − 0.782i)23-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + (0.797 − 0.460i)5-s + (0.109 + 0.993i)7-s − 0.353·8-s + (0.563 + 0.325i)10-s + (−0.390 − 0.920i)11-s − 1.93i·13-s + (−0.569 + 0.418i)14-s + (−0.125 − 0.216i)16-s + (0.646 − 1.11i)17-s + (1.69 − 0.975i)19-s + 0.460i·20-s + (0.425 − 0.564i)22-s + (0.282 − 0.163i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0153i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0153i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1386\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 11\)
Sign: $0.999 - 0.0153i$
Analytic conductor: \(11.0672\)
Root analytic conductor: \(3.32675\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1386} (1187, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1386,\ (\ :1/2),\ 0.999 - 0.0153i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.247012745\)
\(L(\frac12)\) \(\approx\) \(2.247012745\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 - 0.866i)T \)
3 \( 1 \)
7 \( 1 + (-0.289 - 2.62i)T \)
11 \( 1 + (1.29 + 3.05i)T \)
good5 \( 1 + (-1.78 + 1.02i)T + (2.5 - 4.33i)T^{2} \)
13 \( 1 + 6.97iT - 13T^{2} \)
17 \( 1 + (-2.66 + 4.61i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-7.36 + 4.25i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.35 + 0.782i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 2.87T + 29T^{2} \)
31 \( 1 + (4.39 - 7.60i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.82 + 3.15i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 2.42T + 41T^{2} \)
43 \( 1 - 7.55iT - 43T^{2} \)
47 \( 1 + (-10.2 + 5.91i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-9.93 - 5.73i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (3.49 + 2.01i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (4.39 - 2.53i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.75 - 8.23i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 3.30iT - 71T^{2} \)
73 \( 1 + (0.828 + 0.478i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-7.75 + 4.47i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 - 7.50T + 83T^{2} \)
89 \( 1 + (2.99 - 1.72i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 5.95T + 97T^{2} \)
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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

−9.230394835354384418694620166489, −8.911374106600069702817151663777, −7.84304256478159630009966231807, −7.25667125856264196271061883671, −5.85965630074456187420213794082, −5.38394087956813176598343283987, −5.14414462682663292563827662742, −3.25325562337665401037467787056, −2.74731060147287705140268349413, −0.887839984408539186469422909703, 1.47053954395859763408956408482, 2.18413893278357396651680572064, 3.64037352152433920334117845915, 4.22772464002280619457869599704, 5.36168588172451309607529806023, 6.18170343759415376664236769648, 7.14918030097171270375593358908, 7.75121186787797791508246060789, 9.180025042045472518837383687336, 9.768814385881037993612890730357

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