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Properties

Label 2-1386-21.5-c1-0-0
Degree $2$
Conductor $1386$
Sign $-0.958 - 0.286i$
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  + (−0.866 − 0.5i)2-s + (0.499 + 0.866i)4-s + (−0.741 + 1.28i)5-s + (−1.48 + 2.19i)7-s − 0.999i·8-s + (1.28 − 0.741i)10-s + (−0.866 + 0.5i)11-s − 2.44i·13-s + (2.38 − 1.15i)14-s + (−0.5 + 0.866i)16-s + (0.182 + 0.315i)17-s + (7.06 + 4.07i)19-s − 1.48·20-s + 0.999·22-s + (−5.18 − 2.99i)23-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (0.249 + 0.433i)4-s + (−0.331 + 0.574i)5-s + (−0.560 + 0.827i)7-s − 0.353i·8-s + (0.405 − 0.234i)10-s + (−0.261 + 0.150i)11-s − 0.679i·13-s + (0.636 − 0.308i)14-s + (−0.125 + 0.216i)16-s + (0.0441 + 0.0765i)17-s + (1.62 + 0.935i)19-s − 0.331·20-s + 0.213·22-s + (−1.08 − 0.623i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.958 - 0.286i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.958 - 0.286i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1386\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 11\)
Sign: $-0.958 - 0.286i$
Analytic conductor: \(11.0672\)
Root analytic conductor: \(3.32675\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1386} (89, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1386,\ (\ :1/2),\ -0.958 - 0.286i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3108517175\)
\(L(\frac12)\) \(\approx\) \(0.3108517175\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 + 0.5i)T \)
3 \( 1 \)
7 \( 1 + (1.48 - 2.19i)T \)
11 \( 1 + (0.866 - 0.5i)T \)
good5 \( 1 + (0.741 - 1.28i)T + (-2.5 - 4.33i)T^{2} \)
13 \( 1 + 2.44iT - 13T^{2} \)
17 \( 1 + (-0.182 - 0.315i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-7.06 - 4.07i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (5.18 + 2.99i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 0.332iT - 29T^{2} \)
31 \( 1 + (-0.752 + 0.434i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (4.64 - 8.04i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 9.38T + 41T^{2} \)
43 \( 1 + 2.94T + 43T^{2} \)
47 \( 1 + (-0.637 + 1.10i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (4.65 - 2.68i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (4.43 + 7.68i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (7.28 + 4.20i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.08 + 8.79i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 2.07iT - 71T^{2} \)
73 \( 1 + (7.86 - 4.54i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-3.00 + 5.21i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 15.5T + 83T^{2} \)
89 \( 1 + (0.459 - 0.796i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 1.79iT - 97T^{2} \)
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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

−10.04797371130502471584119538949, −9.246142859425524421294308540565, −8.278849876792236140204855045715, −7.73462512592260804226546259921, −6.78047683909583596592693471392, −5.93667290060789289507711637297, −4.97703225054994633582614200044, −3.41072408361566238809581926166, −3.03517040048497318515829138054, −1.68714802142730609386098278770, 0.15885070772451426159360738867, 1.42716239117819610111639915022, 3.01293022035422250028995847586, 4.11004288529463274242312793541, 5.05207150325202309592214103316, 6.01436861916065314600753763930, 7.06100573597245721450628438156, 7.47640603678470370244557796365, 8.451273215722897594654527195745, 9.192308143856487016530120048837

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