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Properties

Label 2-1386-33.8-c1-0-11
Degree $2$
Conductor $1386$
Sign $0.982 + 0.188i$
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 + (−2.53 + 3.48i)5-s + (0.951 + 0.309i)7-s + (0.309 + 0.951i)8-s − 4.31i·10-s + (2.59 − 2.06i)11-s + (−3.41 − 4.69i)13-s + (−0.951 + 0.309i)14-s + (−0.809 − 0.587i)16-s + (−3.36 − 2.44i)17-s + (2.74 − 0.890i)19-s + (2.53 + 3.48i)20-s + (−0.891 + 3.19i)22-s + 5.92i·23-s + ⋯
L(s)  = 1  + (−0.572 + 0.415i)2-s + (0.154 − 0.475i)4-s + (−1.13 + 1.55i)5-s + (0.359 + 0.116i)7-s + (0.109 + 0.336i)8-s − 1.36i·10-s + (0.783 − 0.621i)11-s + (−0.946 − 1.30i)13-s + (−0.254 + 0.0825i)14-s + (−0.202 − 0.146i)16-s + (−0.815 − 0.592i)17-s + (0.628 − 0.204i)19-s + (0.566 + 0.779i)20-s + (−0.190 + 0.681i)22-s + 1.23i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.982 + 0.188i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.982 + 0.188i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1386\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 11\)
Sign: $0.982 + 0.188i$
Analytic conductor: \(11.0672\)
Root analytic conductor: \(3.32675\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1386} (701, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1386,\ (\ :1/2),\ 0.982 + 0.188i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7967582632\)
\(L(\frac12)\) \(\approx\) \(0.7967582632\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.59 + 2.06i)T \)
good5 \( 1 + (2.53 - 3.48i)T + (-1.54 - 4.75i)T^{2} \)
13 \( 1 + (3.41 + 4.69i)T + (-4.01 + 12.3i)T^{2} \)
17 \( 1 + (3.36 + 2.44i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-2.74 + 0.890i)T + (15.3 - 11.1i)T^{2} \)
23 \( 1 - 5.92iT - 23T^{2} \)
29 \( 1 + (-2.53 + 7.79i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-5.02 + 3.64i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (2.74 - 8.43i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (1.31 + 4.03i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 2.81iT - 43T^{2} \)
47 \( 1 + (-3.74 + 1.21i)T + (38.0 - 27.6i)T^{2} \)
53 \( 1 + (-5.28 - 7.27i)T + (-16.3 + 50.4i)T^{2} \)
59 \( 1 + (6.92 + 2.25i)T + (47.7 + 34.6i)T^{2} \)
61 \( 1 + (-0.553 + 0.762i)T + (-18.8 - 58.0i)T^{2} \)
67 \( 1 - 15.2T + 67T^{2} \)
71 \( 1 + (-0.423 + 0.583i)T + (-21.9 - 67.5i)T^{2} \)
73 \( 1 + (5.71 + 1.85i)T + (59.0 + 42.9i)T^{2} \)
79 \( 1 + (-8.25 - 11.3i)T + (-24.4 + 75.1i)T^{2} \)
83 \( 1 + (4.38 + 3.18i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + 4.27iT - 89T^{2} \)
97 \( 1 + (0.399 - 0.290i)T + (29.9 - 92.2i)T^{2} \)
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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

−9.677065913882955848649765050419, −8.487574050209674563478664628812, −7.81844982769208934715005474850, −7.26028443237991752427384981437, −6.54628707286640859131532057547, −5.60975643291780946737978268779, −4.40420348972908192057181446765, −3.29288018865604035359036269385, −2.51041091036153087110599279359, −0.49165197122340025644012597185, 1.02640331387712662022807244075, 2.04099341547949767443189745469, 3.73331970757859800362279784575, 4.50160264535646185136662076884, 4.95010920890968413994208280285, 6.65731018971648706693317232940, 7.33296885849425126458818087244, 8.233635054623253773944357845999, 8.895649764017066832833350796298, 9.277644833018357621002844412150

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