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Properties

Label 2-1386-7.4-c1-0-3
Degree $2$
Conductor $1386$
Sign $-0.0725 - 0.997i$
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 + (−1.70 − 2.95i)5-s + (−2.62 + 0.358i)7-s − 0.999·8-s + (1.70 − 2.95i)10-s + (0.5 − 0.866i)11-s + 1.82·13-s + (−1.62 − 2.09i)14-s + (−0.5 − 0.866i)16-s + (−3.82 + 6.63i)17-s + (1.70 + 2.95i)19-s + 3.41·20-s + 0.999·22-s + (1.12 + 1.94i)23-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + (−0.763 − 1.32i)5-s + (−0.990 + 0.135i)7-s − 0.353·8-s + (0.539 − 0.935i)10-s + (0.150 − 0.261i)11-s + 0.507·13-s + (−0.433 − 0.558i)14-s + (−0.125 − 0.216i)16-s + (−0.928 + 1.60i)17-s + (0.391 + 0.678i)19-s + 0.763·20-s + 0.213·22-s + (0.233 + 0.404i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0725 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0725 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1386\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 11\)
Sign: $-0.0725 - 0.997i$
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: \((2,\ 1386,\ (\ :1/2),\ -0.0725 - 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.153080900\)
\(L(\frac12)\) \(\approx\) \(1.153080900\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.62 - 0.358i)T \)
11 \( 1 + (-0.5 + 0.866i)T \)
good5 \( 1 + (1.70 + 2.95i)T + (-2.5 + 4.33i)T^{2} \)
13 \( 1 - 1.82T + 13T^{2} \)
17 \( 1 + (3.82 - 6.63i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.70 - 2.95i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.12 - 1.94i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 8.65T + 29T^{2} \)
31 \( 1 + (-2 + 3.46i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-3.29 - 5.70i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 2.58T + 41T^{2} \)
43 \( 1 - 5.65T + 43T^{2} \)
47 \( 1 + (-3.24 - 5.61i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (5.94 - 10.3i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (4.20 - 7.28i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.08 + 5.34i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.62 - 9.73i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 3.07T + 71T^{2} \)
73 \( 1 + (-3.29 + 5.70i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2.37 - 4.11i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 16.1T + 83T^{2} \)
89 \( 1 + (2.24 + 3.88i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 1.82T + 97T^{2} \)
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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

−9.495906644236889521013785242040, −8.731391326506155723710740219683, −8.285686379775834893414984967023, −7.43011093025484928385288641421, −6.20405512871942641918250070681, −5.90730961329501891166193816931, −4.52770823154717838700104514266, −4.09586248631227615082460066769, −3.02381880984752048594946359772, −1.13235652129282728469954378969, 0.49158167002211527605023199551, 2.59587017626661169940388430426, 3.04187915196937989360544371497, 4.01575462380281164272132198203, 4.91187433735736398387896429515, 6.31169845196690366901169820586, 6.81841294357863948575366180259, 7.48933789252474001631605144195, 8.781892845747340239423792698563, 9.516367604289941910145732344039

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