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Properties

Label 2-1386-7.4-c1-0-0
Degree $2$
Conductor $1386$
Sign $-0.969 + 0.243i$
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 + (−0.806 − 1.39i)5-s + (−0.806 − 2.51i)7-s − 0.999·8-s + (0.806 − 1.39i)10-s + (−0.5 + 0.866i)11-s + (1.77 − 1.95i)14-s + (−0.5 − 0.866i)16-s + (−3.67 + 6.35i)17-s + (−0.585 − 1.01i)19-s + 1.61·20-s − 0.999·22-s + (1.77 + 3.08i)23-s + (1.19 − 2.07i)25-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + (−0.360 − 0.624i)5-s + (−0.304 − 0.952i)7-s − 0.353·8-s + (0.255 − 0.441i)10-s + (−0.150 + 0.261i)11-s + (0.475 − 0.523i)14-s + (−0.125 − 0.216i)16-s + (−0.890 + 1.54i)17-s + (−0.134 − 0.232i)19-s + 0.360·20-s − 0.213·22-s + (0.370 + 0.642i)23-s + (0.239 − 0.415i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.969 + 0.243i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.969 + 0.243i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1386\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 11\)
Sign: $-0.969 + 0.243i$
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: \((2,\ 1386,\ (\ :1/2),\ -0.969 + 0.243i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1540408325\)
\(L(\frac12)\) \(\approx\) \(0.1540408325\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.806 + 2.51i)T \)
11 \( 1 + (0.5 - 0.866i)T \)
good5 \( 1 + (0.806 + 1.39i)T + (-2.5 + 4.33i)T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 + (3.67 - 6.35i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.585 + 1.01i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.77 - 3.08i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 10.3T + 29T^{2} \)
31 \( 1 + (3.17 - 5.49i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.41 + 2.44i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 4.94T + 41T^{2} \)
43 \( 1 + 11.5T + 43T^{2} \)
47 \( 1 + (-3.39 - 5.87i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-4 + 6.92i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2.19 + 3.80i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5.97 - 10.3i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.47 - 2.55i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 15.5T + 71T^{2} \)
73 \( 1 + (-2.55 + 4.43i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (2.80 + 4.86i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 5.05T + 83T^{2} \)
89 \( 1 + (0.386 + 0.669i)T + (-44.5 + 77.0i)T^{2} \)
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

−9.987817749953366180399760332882, −8.920880639857556516268206271418, −8.405435466499892114770841779470, −7.41224276257706113846018501648, −6.87665765136814148073249558345, −5.89388923723393740722906772120, −4.92734558074786769079285904895, −4.11707555782188481031278959576, −3.43503201483177406835474677996, −1.69678511467731903539240390777, 0.05245301872664529235129904413, 2.05481920505225074586457652263, 2.90592346441474990881941689133, 3.71902958085182317952577649897, 4.92936932533881873615008471483, 5.64620513729771702434035436347, 6.66912680789665113995581477582, 7.37932837093462104112597578607, 8.592073388776984794999622514599, 9.203154882968182270836067025102

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