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Properties

Label 2-1386-7.4-c1-0-14
Degree $2$
Conductor $1386$
Sign $0.895 - 0.444i$
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  + (−0.5 − 0.866i)2-s + (−0.499 + 0.866i)4-s + (1.82 + 3.15i)5-s + (1.32 + 2.29i)7-s + 0.999·8-s + (1.82 − 3.15i)10-s + (0.5 − 0.866i)11-s + 5·13-s + (1.32 − 2.29i)14-s + (−0.5 − 0.866i)16-s + (3 − 5.19i)17-s + (0.177 + 0.306i)19-s − 3.64·20-s − 0.999·22-s + (−1.82 − 3.15i)23-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s + (0.815 + 1.41i)5-s + (0.499 + 0.866i)7-s + 0.353·8-s + (0.576 − 0.998i)10-s + (0.150 − 0.261i)11-s + 1.38·13-s + (0.353 − 0.612i)14-s + (−0.125 − 0.216i)16-s + (0.727 − 1.26i)17-s + (0.0406 + 0.0703i)19-s − 0.815·20-s − 0.213·22-s + (−0.380 − 0.658i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.895 - 0.444i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.895 - 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1386\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 11\)
Sign: $0.895 - 0.444i$
Analytic conductor: \(11.0672\)
Root analytic conductor: \(3.32675\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1386} (991, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1386,\ (\ :1/2),\ 0.895 - 0.444i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.856829341\)
\(L(\frac12)\) \(\approx\) \(1.856829341\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 + 0.866i)T \)
3 \( 1 \)
7 \( 1 + (-1.32 - 2.29i)T \)
11 \( 1 + (-0.5 + 0.866i)T \)
good5 \( 1 + (-1.82 - 3.15i)T + (-2.5 + 4.33i)T^{2} \)
13 \( 1 - 5T + 13T^{2} \)
17 \( 1 + (-3 + 5.19i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.177 - 0.306i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.82 + 3.15i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 4.29T + 29T^{2} \)
31 \( 1 + (-2 + 3.46i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.822 - 1.42i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 4.93T + 41T^{2} \)
43 \( 1 + 4T + 43T^{2} \)
47 \( 1 + (-6.64 - 11.5i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.82 - 3.15i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.322 - 0.559i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.85 + 3.21i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.96 - 3.40i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 9.64T + 71T^{2} \)
73 \( 1 + (2.82 - 4.88i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (1.32 + 2.29i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 13.2T + 83T^{2} \)
89 \( 1 + (7.29 + 12.6i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 5.70T + 97T^{2} \)
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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

−9.739504207606365379442345235118, −8.977604773230659414681108390806, −8.173663245726444893825739921025, −7.25719850835924045382920438227, −6.20725411230432128902852773991, −5.74282989916066471022986272219, −4.39447154289855556606540193219, −3.05542150373545098341434143491, −2.61898972939481430675837603024, −1.36293796693553472806189891109, 1.06439894563318349657540342142, 1.64985462577061168119292547808, 3.76559883339007094640113604820, 4.55867703494728280491804111965, 5.51472189663691442290236474300, 6.08856887979021775552599421224, 7.11863415451105263551688547735, 8.272306512017935872376760583755, 8.439619227749878678783003884250, 9.399662279692689686578321045059

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