Properties

Label 2-1386-7.2-c1-0-31
Degree $2$
Conductor $1386$
Sign $-0.386 - 0.922i$
Analytic cond. $11.0672$
Root an. cond. $3.32675$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 − 0.866i)2-s + (−0.499 − 0.866i)4-s + (1 − 1.73i)5-s + (−2.5 + 0.866i)7-s − 0.999·8-s + (−0.999 − 1.73i)10-s + (−0.5 − 0.866i)11-s − 7·13-s + (−0.500 + 2.59i)14-s + (−0.5 + 0.866i)16-s + (1 + 1.73i)17-s − 1.99·20-s − 0.999·22-s + (−4 + 6.92i)23-s + (0.500 + 0.866i)25-s + (−3.5 + 6.06i)26-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (0.447 − 0.774i)5-s + (−0.944 + 0.327i)7-s − 0.353·8-s + (−0.316 − 0.547i)10-s + (−0.150 − 0.261i)11-s − 1.94·13-s + (−0.133 + 0.694i)14-s + (−0.125 + 0.216i)16-s + (0.242 + 0.420i)17-s − 0.447·20-s − 0.213·22-s + (−0.834 + 1.44i)23-s + (0.100 + 0.173i)25-s + (−0.686 + 1.18i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.386 - 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1386\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 11\)
Sign: $-0.386 - 0.922i$
Analytic conductor: \(11.0672\)
Root analytic conductor: \(3.32675\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1386} (793, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 1386,\ (\ :1/2),\ -0.386 - 0.922i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.5 - 0.866i)T \)
11 \( 1 + (0.5 + 0.866i)T \)
good5 \( 1 + (-1 + 1.73i)T + (-2.5 - 4.33i)T^{2} \)
13 \( 1 + 7T + 13T^{2} \)
17 \( 1 + (-1 - 1.73i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (4 - 6.92i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 5T + 29T^{2} \)
31 \( 1 + (2 + 3.46i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (2 - 3.46i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 4T + 41T^{2} \)
43 \( 1 + 8T + 43T^{2} \)
47 \( 1 + (-1 + 1.73i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (3 + 5.19i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-1.5 - 2.59i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.5 + 7.79i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 2T + 71T^{2} \)
73 \( 1 + (2 + 3.46i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (4.5 - 7.79i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 6T + 83T^{2} \)
89 \( 1 + (-3 + 5.19i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 7T + 97T^{2} \)
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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

−9.347146940890400031167254519453, −8.402711167269090113813159075923, −7.38053985947741429936379499473, −6.35447981703526983183437500642, −5.43014031734927876090125655462, −4.91386442592581204627499716929, −3.68612678543057485420691162952, −2.73874047689079527364115532002, −1.67638202708494261981625072922, 0, 2.41477780911271363056326392541, 3.06702934538345590599517576196, 4.34586621938657870757983730193, 5.15038466116120683249202141835, 6.22151945108275359559471990606, 6.87195576413003465722387076695, 7.35142333821652578905366075368, 8.393748599256950342727763959778, 9.463651737599295965675675668980

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