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Properties

Label 2-1386-77.54-c1-0-0
Degree $2$
Conductor $1386$
Sign $-0.674 - 0.738i$
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 + (−0.882 − 0.509i)5-s + (−2.25 − 1.38i)7-s − 0.999i·8-s + (0.509 + 0.882i)10-s + (2.58 + 2.08i)11-s + 0.167·13-s + (1.26 + 2.32i)14-s + (−0.5 + 0.866i)16-s + (−1.47 − 2.55i)17-s + (0.155 − 0.269i)19-s − 1.01i·20-s + (−1.19 − 3.09i)22-s + (−0.237 + 0.411i)23-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (0.249 + 0.433i)4-s + (−0.394 − 0.227i)5-s + (−0.852 − 0.522i)7-s − 0.353i·8-s + (0.161 + 0.279i)10-s + (0.778 + 0.627i)11-s + 0.0463·13-s + (0.337 + 0.621i)14-s + (−0.125 + 0.216i)16-s + (−0.358 − 0.620i)17-s + (0.0357 − 0.0619i)19-s − 0.227i·20-s + (−0.255 − 0.659i)22-s + (−0.0495 + 0.0858i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.674 - 0.738i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.674 - 0.738i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1386\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 11\)
Sign: $-0.674 - 0.738i$
Analytic conductor: \(11.0672\)
Root analytic conductor: \(3.32675\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1386} (901, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1386,\ (\ :1/2),\ -0.674 - 0.738i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.09588520237\)
\(L(\frac12)\) \(\approx\) \(0.09588520237\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.25 + 1.38i)T \)
11 \( 1 + (-2.58 - 2.08i)T \)
good5 \( 1 + (0.882 + 0.509i)T + (2.5 + 4.33i)T^{2} \)
13 \( 1 - 0.167T + 13T^{2} \)
17 \( 1 + (1.47 + 2.55i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.155 + 0.269i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.237 - 0.411i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 1.89iT - 29T^{2} \)
31 \( 1 + (2.20 - 1.27i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (3.04 - 5.27i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 10.2T + 41T^{2} \)
43 \( 1 + 10.1iT - 43T^{2} \)
47 \( 1 + (-3.28 - 1.89i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-4.21 - 7.29i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (5.40 - 3.12i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.93 - 10.2i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.19 - 8.99i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 14.5T + 71T^{2} \)
73 \( 1 + (-4.85 - 8.40i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (6.06 + 3.50i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 14.6T + 83T^{2} \)
89 \( 1 + (5.97 + 3.45i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 11.0iT - 97T^{2} \)
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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

−9.988358669616763605930077256810, −9.022709983221617429665291614311, −8.539725587406852405587361489122, −7.20055670086107096459449536686, −7.05638700542666824315403975679, −5.89350166799199452870561797982, −4.53895865439054948668098265553, −3.79938022671804183573111361752, −2.77051161288721557863756871091, −1.38176287651468527127481746881, 0.04915356416730803387085863729, 1.75533266630029251300946315734, 3.13606658750468621499649482242, 3.95422092613391825506769309235, 5.35437008158217544515882733459, 6.22600448792068611713536126416, 6.74594767756083194229830111962, 7.73732535532015424481215110557, 8.553948035336614896972228938843, 9.209363072742069357624138494265

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