Properties

Label 2-1386-33.8-c1-0-22
Degree $2$
Conductor $1386$
Sign $-0.505 + 0.862i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.505 + 0.862i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.505 + 0.862i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1386\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 11\)
Sign: $-0.505 + 0.862i$
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: \((2,\ 1386,\ (\ :1/2),\ -0.505 + 0.862i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.664001810\)
\(L(\frac12)\) \(\approx\) \(2.664001810\)
\(L(\frac{3}{2})\) 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

−9.681824327033003009081261253559, −8.426831880121790820952789713055, −7.966588735485730598302193780133, −6.62819914529451631694473112214, −5.38157907762686649774444142959, −5.27388864321639990611194034357, −4.50279204635413481875882145513, −2.95533332135039462995735138841, −1.99515688899690059393538978912, −0.889712007630160244523618186905, 1.98496469686018486322574889887, 2.80129214112550343058952120800, 3.71005398808492782577254381962, 5.10749773407808500571598950283, 5.70005766338117819583160165362, 6.53047613675164054467794266531, 7.34588030734590467889374939973, 7.77466734723764475142007609133, 9.283107415855681987756499402953, 9.811935946557355928607186982986

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