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Properties

Label 2-1386-33.17-c1-0-6
Degree $2$
Conductor $1386$
Sign $-0.320 - 0.947i$
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  + (0.309 + 0.951i)2-s + (−0.809 + 0.587i)4-s + (−4.02 − 1.30i)5-s + (0.587 + 0.809i)7-s + (−0.809 − 0.587i)8-s − 4.23i·10-s + (2.00 − 2.63i)11-s + (−0.133 + 0.0432i)13-s + (−0.587 + 0.809i)14-s + (0.309 − 0.951i)16-s + (0.224 − 0.692i)17-s + (3.29 − 4.53i)19-s + (4.02 − 1.30i)20-s + (3.13 + 1.09i)22-s + 8.26i·23-s + ⋯
L(s)  = 1  + (0.218 + 0.672i)2-s + (−0.404 + 0.293i)4-s + (−1.80 − 0.585i)5-s + (0.222 + 0.305i)7-s + (−0.286 − 0.207i)8-s − 1.33i·10-s + (0.605 − 0.795i)11-s + (−0.0369 + 0.0119i)13-s + (−0.157 + 0.216i)14-s + (0.0772 − 0.237i)16-s + (0.0545 − 0.167i)17-s + (0.756 − 1.04i)19-s + (0.900 − 0.292i)20-s + (0.667 + 0.233i)22-s + 1.72i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.320 - 0.947i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.320 - 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1386\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 11\)
Sign: $-0.320 - 0.947i$
Analytic conductor: \(11.0672\)
Root analytic conductor: \(3.32675\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1386} (1205, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1386,\ (\ :1/2),\ -0.320 - 0.947i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9784036925\)
\(L(\frac12)\) \(\approx\) \(0.9784036925\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.309 - 0.951i)T \)
3 \( 1 \)
7 \( 1 + (-0.587 - 0.809i)T \)
11 \( 1 + (-2.00 + 2.63i)T \)
good5 \( 1 + (4.02 + 1.30i)T + (4.04 + 2.93i)T^{2} \)
13 \( 1 + (0.133 - 0.0432i)T + (10.5 - 7.64i)T^{2} \)
17 \( 1 + (-0.224 + 0.692i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (-3.29 + 4.53i)T + (-5.87 - 18.0i)T^{2} \)
23 \( 1 - 8.26iT - 23T^{2} \)
29 \( 1 + (6.86 - 4.98i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-3.04 - 9.38i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (8.44 - 6.13i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (3.05 + 2.22i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 0.646iT - 43T^{2} \)
47 \( 1 + (-6.10 + 8.40i)T + (-14.5 - 44.6i)T^{2} \)
53 \( 1 + (2.86 - 0.932i)T + (42.8 - 31.1i)T^{2} \)
59 \( 1 + (-0.186 - 0.256i)T + (-18.2 + 56.1i)T^{2} \)
61 \( 1 + (-5.69 - 1.84i)T + (49.3 + 35.8i)T^{2} \)
67 \( 1 - 4.47T + 67T^{2} \)
71 \( 1 + (-5.39 - 1.75i)T + (57.4 + 41.7i)T^{2} \)
73 \( 1 + (-8.32 - 11.4i)T + (-22.5 + 69.4i)T^{2} \)
79 \( 1 + (2.12 - 0.689i)T + (63.9 - 46.4i)T^{2} \)
83 \( 1 + (1.04 - 3.21i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 - 8.13iT - 89T^{2} \)
97 \( 1 + (-2.96 - 9.12i)T + (-78.4 + 57.0i)T^{2} \)
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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

−9.413544977059637074675561623947, −8.714880740406608891657609477778, −8.275232856403724226797954850017, −7.26139354675622926503790612624, −6.91049594042353988925224256162, −5.34553695587611839226241301521, −5.00336043212491828367511846954, −3.71040112641225490121706108040, −3.34721837609758501509039876313, −1.07629058405632862621683992988, 0.46592383624529255118841635187, 2.12796553490577403614963326414, 3.41262493776306500447210362492, 4.05779962949876015607843411961, 4.63586815745463151524772283150, 6.04556746552168933858863536803, 7.08581370683828264020114737494, 7.73735932534716272756398198192, 8.394898501497793942285298323288, 9.500341438532740747560976205906

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