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Properties

Label 2-1386-77.10-c1-0-0
Degree $2$
Conductor $1386$
Sign $0.325 + 0.945i$
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 + (−3.71 + 2.14i)5-s + (−0.293 + 2.62i)7-s + 0.999i·8-s + (2.14 − 3.71i)10-s + (−2.21 + 2.46i)11-s − 4.87·13-s + (−1.06 − 2.42i)14-s + (−0.5 − 0.866i)16-s + (−1.58 + 2.75i)17-s + (−1.05 − 1.82i)19-s + 4.28i·20-s + (0.682 − 3.24i)22-s + (−1.06 − 1.83i)23-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (0.249 − 0.433i)4-s + (−1.66 + 0.958i)5-s + (−0.110 + 0.993i)7-s + 0.353i·8-s + (0.677 − 1.17i)10-s + (−0.667 + 0.744i)11-s − 1.35·13-s + (−0.283 − 0.647i)14-s + (−0.125 − 0.216i)16-s + (−0.385 + 0.667i)17-s + (−0.241 − 0.418i)19-s + 0.958i·20-s + (0.145 − 0.691i)22-s + (−0.221 − 0.382i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.325 + 0.945i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.325 + 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1386\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 11\)
Sign: $0.325 + 0.945i$
Analytic conductor: \(11.0672\)
Root analytic conductor: \(3.32675\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1386} (703, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1386,\ (\ :1/2),\ 0.325 + 0.945i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.03906174470\)
\(L(\frac12)\) \(\approx\) \(0.03906174470\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.293 - 2.62i)T \)
11 \( 1 + (2.21 - 2.46i)T \)
good5 \( 1 + (3.71 - 2.14i)T + (2.5 - 4.33i)T^{2} \)
13 \( 1 + 4.87T + 13T^{2} \)
17 \( 1 + (1.58 - 2.75i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.05 + 1.82i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.06 + 1.83i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 4.88iT - 29T^{2} \)
31 \( 1 + (-5.20 - 3.00i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.75 - 3.04i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 5.28T + 41T^{2} \)
43 \( 1 - 9.33iT - 43T^{2} \)
47 \( 1 + (-7.40 + 4.27i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-3.09 + 5.36i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-0.485 - 0.280i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.06 + 1.84i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.94 + 6.82i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 9.03T + 71T^{2} \)
73 \( 1 + (0.587 - 1.01i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-10.1 + 5.83i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 14.1T + 83T^{2} \)
89 \( 1 + (1.01 - 0.587i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 10.6iT - 97T^{2} \)
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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

−10.27249050152084577151418927644, −9.362858674557746987318191103302, −8.278710205551700548265336722271, −7.975416440172708452142724559157, −6.95893121854102075526252773974, −6.60707415261544762406384173505, −5.17278169282678079698385667922, −4.40178850536539229080877743723, −3.04723096309934855532435184894, −2.30659487781539774609975573354, 0.02780204159458473129822221464, 0.804775308038717473579250917604, 2.62065922856056512976682285643, 3.80089518498167929895106624856, 4.39182711694740922683392359005, 5.36276657254089547183772935277, 6.90954433084820568997566656390, 7.64932638787348649066725346598, 7.995585595919748436384277920748, 8.847559797969785513188472981221

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