Properties

Degree $2$
Conductor $1386$
Sign $0.761 - 0.648i$
Motivic weight $1$
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 + (2.34 + 1.35i)5-s + (−0.222 − 2.63i)7-s + 0.999·8-s + (−2.34 + 1.35i)10-s + (−2.96 + 1.48i)11-s − 2.18i·13-s + (2.39 + 1.12i)14-s + (−0.5 + 0.866i)16-s + (0.0281 + 0.0488i)17-s + (2.38 + 1.37i)19-s − 2.71i·20-s + (0.192 − 3.31i)22-s + (6.02 + 3.47i)23-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + (1.05 + 0.606i)5-s + (−0.0839 − 0.996i)7-s + 0.353·8-s + (−0.742 + 0.428i)10-s + (−0.893 + 0.449i)11-s − 0.604i·13-s + (0.639 + 0.300i)14-s + (−0.125 + 0.216i)16-s + (0.00683 + 0.0118i)17-s + (0.547 + 0.315i)19-s − 0.606i·20-s + (0.0409 − 0.705i)22-s + (1.25 + 0.725i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.761 - 0.648i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.761 - 0.648i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1386\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 11\)
Sign: $0.761 - 0.648i$
Motivic weight: \(1\)
Character: $\chi_{1386} (989, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1386,\ (\ :1/2),\ 0.761 - 0.648i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.610164094\)
\(L(\frac12)\) \(\approx\) \(1.610164094\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.222 + 2.63i)T \)
11 \( 1 + (2.96 - 1.48i)T \)
good5 \( 1 + (-2.34 - 1.35i)T + (2.5 + 4.33i)T^{2} \)
13 \( 1 + 2.18iT - 13T^{2} \)
17 \( 1 + (-0.0281 - 0.0488i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.38 - 1.37i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-6.02 - 3.47i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 6.43T + 29T^{2} \)
31 \( 1 + (-3.65 - 6.32i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.62 - 2.80i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 2.71T + 41T^{2} \)
43 \( 1 + 10.6iT - 43T^{2} \)
47 \( 1 + (2.58 + 1.49i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-10.1 + 5.88i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-12.2 + 7.07i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-10.2 - 5.93i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.749 - 1.29i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 1.82iT - 71T^{2} \)
73 \( 1 + (-6.82 + 3.93i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (7.82 + 4.51i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 17.2T + 83T^{2} \)
89 \( 1 + (-7.43 - 4.29i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 6.25T + 97T^{2} \)
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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

−10.04754067288616354279467564096, −8.807649884828350265437602569174, −7.948990882290359977275829523238, −7.06334000890482704754300731648, −6.67623442351101196475295155019, −5.49449222012323866471221351672, −4.98214157478853749058524109681, −3.54128377885654586738912451010, −2.44762449211278752993532422678, −1.02083936834312697531591828925, 0.996821314437607389598688705367, 2.33320271599694106575082271748, 2.88373654023682187253630300022, 4.48090498228671683719639309761, 5.30744327923881727320996852019, 6.00281094208054321847429392642, 7.06649656065206068177480528633, 8.332288835901006481521617369824, 8.739626123432341946005837499407, 9.581197420901594323670136384112

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