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Properties

Label 2-1386-33.2-c1-0-5
Degree $2$
Conductor $1386$
Sign $0.974 - 0.223i$
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 + (−1.81 + 0.591i)5-s + (0.587 − 0.809i)7-s + (−0.809 + 0.587i)8-s + 1.91i·10-s + (2.51 + 2.15i)11-s + (−2.40 − 0.781i)13-s + (−0.587 − 0.809i)14-s + (0.309 + 0.951i)16-s + (1.61 + 4.96i)17-s + (−1.99 − 2.73i)19-s + (1.81 + 0.591i)20-s + (2.82 − 1.72i)22-s − 0.978i·23-s + ⋯
L(s)  = 1  + (0.218 − 0.672i)2-s + (−0.404 − 0.293i)4-s + (−0.813 + 0.264i)5-s + (0.222 − 0.305i)7-s + (−0.286 + 0.207i)8-s + 0.604i·10-s + (0.759 + 0.650i)11-s + (−0.666 − 0.216i)13-s + (−0.157 − 0.216i)14-s + (0.0772 + 0.237i)16-s + (0.391 + 1.20i)17-s + (−0.456 − 0.628i)19-s + (0.406 + 0.132i)20-s + (0.603 − 0.368i)22-s − 0.204i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.974 - 0.223i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.974 - 0.223i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1386\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 11\)
Sign: $0.974 - 0.223i$
Analytic conductor: \(11.0672\)
Root analytic conductor: \(3.32675\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1386} (827, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1386,\ (\ :1/2),\ 0.974 - 0.223i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.352560446\)
\(L(\frac12)\) \(\approx\) \(1.352560446\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.51 - 2.15i)T \)
good5 \( 1 + (1.81 - 0.591i)T + (4.04 - 2.93i)T^{2} \)
13 \( 1 + (2.40 + 0.781i)T + (10.5 + 7.64i)T^{2} \)
17 \( 1 + (-1.61 - 4.96i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (1.99 + 2.73i)T + (-5.87 + 18.0i)T^{2} \)
23 \( 1 + 0.978iT - 23T^{2} \)
29 \( 1 + (-6.64 - 4.82i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (1.58 - 4.87i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-3.15 - 2.29i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (-3.37 + 2.45i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 - 3.26iT - 43T^{2} \)
47 \( 1 + (-2.86 - 3.94i)T + (-14.5 + 44.6i)T^{2} \)
53 \( 1 + (-11.8 - 3.84i)T + (42.8 + 31.1i)T^{2} \)
59 \( 1 + (-2.71 + 3.73i)T + (-18.2 - 56.1i)T^{2} \)
61 \( 1 + (-0.970 + 0.315i)T + (49.3 - 35.8i)T^{2} \)
67 \( 1 - 1.31T + 67T^{2} \)
71 \( 1 + (4.69 - 1.52i)T + (57.4 - 41.7i)T^{2} \)
73 \( 1 + (-2.95 + 4.06i)T + (-22.5 - 69.4i)T^{2} \)
79 \( 1 + (-12.9 - 4.21i)T + (63.9 + 46.4i)T^{2} \)
83 \( 1 + (-1.38 - 4.26i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 - 5.41iT - 89T^{2} \)
97 \( 1 + (-0.223 + 0.686i)T + (-78.4 - 57.0i)T^{2} \)
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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

−9.757482601628496607720345892004, −8.853385876578296245251962850440, −8.042595200877447708834177721903, −7.18495702494256743494184436588, −6.40206112517736743127632785212, −5.14070230349246036245280362801, −4.29707749188477775749538087891, −3.63104972466605828918469268263, −2.48784577495100984683370558100, −1.17826245280873415542867378454, 0.60302403874740806919567082507, 2.49070857886516913109116369373, 3.77889799743589974382907596686, 4.43176761222484891734578750297, 5.42576492821793493356892697493, 6.23666982222131378959689378584, 7.20030365971666862510893914184, 7.88709588251136174624663293905, 8.572752867794865370832823464187, 9.349656929734257152781693518026

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