Properties

Label 2-1386-33.17-c1-0-15
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: $\chi_{1386} (1205, \cdot )$
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

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

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