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Properties

Label 2-1386-7.2-c1-0-30
Degree $2$
Conductor $1386$
Sign $-0.968 + 0.250i$
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 + (1.5 − 2.59i)5-s + (−0.5 − 2.59i)7-s − 0.999·8-s + (−1.5 − 2.59i)10-s + (−0.5 − 0.866i)11-s + 2·13-s + (−2.5 − 0.866i)14-s + (−0.5 + 0.866i)16-s + (−1.5 − 2.59i)17-s + (−1 + 1.73i)19-s − 3·20-s − 0.999·22-s + (1.5 − 2.59i)23-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (0.670 − 1.16i)5-s + (−0.188 − 0.981i)7-s − 0.353·8-s + (−0.474 − 0.821i)10-s + (−0.150 − 0.261i)11-s + 0.554·13-s + (−0.668 − 0.231i)14-s + (−0.125 + 0.216i)16-s + (−0.363 − 0.630i)17-s + (−0.229 + 0.397i)19-s − 0.670·20-s − 0.213·22-s + (0.312 − 0.541i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.968 + 0.250i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.968 + 0.250i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1386\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 11\)
Sign: $-0.968 + 0.250i$
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: \(0\)
Selberg data: \((2,\ 1386,\ (\ :1/2),\ -0.968 + 0.250i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.892391114\)
\(L(\frac12)\) \(\approx\) \(1.892391114\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 + 2.59i)T \)
11 \( 1 + (0.5 + 0.866i)T \)
good5 \( 1 + (-1.5 + 2.59i)T + (-2.5 - 4.33i)T^{2} \)
13 \( 1 - 2T + 13T^{2} \)
17 \( 1 + (1.5 + 2.59i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1 - 1.73i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.5 + 2.59i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 + (1 + 1.73i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (4 - 6.92i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 9T + 41T^{2} \)
43 \( 1 + 4T + 43T^{2} \)
47 \( 1 + (-1.5 + 2.59i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-3 - 5.19i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-3 - 5.19i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.5 - 4.33i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (5.5 + 9.52i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + (1 + 1.73i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-6.5 + 11.2i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 9T + 83T^{2} \)
89 \( 1 + (-6 + 10.3i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 5T + 97T^{2} \)
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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

−9.217062924596311248676893253909, −8.697821537925851870849946695263, −7.66175022574614591161562373478, −6.55230463146272332999433292479, −5.72236823786341076246192069360, −4.81131089079268160735577534770, −4.15138549989586915671663184457, −3.02519036030764978618630354830, −1.68012959672932674705213417658, −0.68325844302784036190369189867, 2.05041342205295863930666685506, 2.91876392780964642375102573307, 3.92243425725762980072636494061, 5.22372394999525014477091871357, 5.92490155316792671113020301215, 6.57120440238752923490264835571, 7.26197844286975942477499885528, 8.333286601262976603405087298477, 9.076758327552317351774415152955, 9.833303178867000007867976614648

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