Properties

Label 2-1386-7.4-c1-0-18
Degree $2$
Conductor $1386$
Sign $0.812 + 0.582i$
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 + (−1.40 − 2.42i)5-s + (−1.40 + 2.24i)7-s − 0.999·8-s + (1.40 − 2.42i)10-s + (−0.5 + 0.866i)11-s + (−2.64 − 0.0932i)14-s + (−0.5 − 0.866i)16-s + (3.98 − 6.89i)17-s + (3.24 + 5.61i)19-s + 2.80·20-s − 0.999·22-s + (−2.64 − 4.57i)23-s + (−1.43 + 2.48i)25-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + (−0.627 − 1.08i)5-s + (−0.530 + 0.847i)7-s − 0.353·8-s + (0.443 − 0.768i)10-s + (−0.150 + 0.261i)11-s + (−0.706 − 0.0249i)14-s + (−0.125 − 0.216i)16-s + (0.965 − 1.67i)17-s + (0.743 + 1.28i)19-s + 0.627·20-s − 0.213·22-s + (−0.551 − 0.954i)23-s + (−0.287 + 0.497i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.812 + 0.582i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.812 + 0.582i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1386\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 11\)
Sign: $0.812 + 0.582i$
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: \((2,\ 1386,\ (\ :1/2),\ 0.812 + 0.582i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.300012139\)
\(L(\frac12)\) \(\approx\) \(1.300012139\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.40 - 2.24i)T \)
11 \( 1 + (0.5 - 0.866i)T \)
good5 \( 1 + (1.40 + 2.42i)T + (-2.5 + 4.33i)T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 + (-3.98 + 6.89i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.24 - 5.61i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.64 + 4.57i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 5.12T + 29T^{2} \)
31 \( 1 + (-4.48 + 7.76i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (5.24 + 9.07i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 5.09T + 41T^{2} \)
43 \( 1 - 11.4T + 43T^{2} \)
47 \( 1 + (-0.161 - 0.279i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-4 + 6.92i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.435 - 0.754i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.07 + 1.87i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.54 + 6.14i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 7.44T + 71T^{2} \)
73 \( 1 + (6.28 - 10.8i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (3.40 + 5.89i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 15.0T + 83T^{2} \)
89 \( 1 + (-0.805 - 1.39i)T + (-44.5 + 77.0i)T^{2} \)
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

−9.401719087190071324390471672872, −8.587001623841608023296877635991, −7.81091275480011354253560736380, −7.23492801299544299444690091938, −5.87721227208616289622225542401, −5.48190391925114917366059480125, −4.49882386466825625984165364425, −3.63040962690912129893661876527, −2.42867313291301469815092619835, −0.53156363397190164673825902769, 1.22623810785327963102929021647, 2.88145781109617046137080430832, 3.47319416524941832648197404654, 4.20194126036395396114437804026, 5.48019534682789116433717095567, 6.39309647048180894641175490957, 7.23189538403720018530700951043, 7.85815120041095997315558257180, 9.003533664392377541230978616845, 10.00750386943822370117123288634

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