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Properties

Label 2-1386-33.8-c1-0-14
Degree $2$
Conductor $1386$
Sign $0.815 + 0.578i$
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  + (−0.809 + 0.587i)2-s + (0.309 − 0.951i)4-s + (2.03 − 2.79i)5-s + (0.951 + 0.309i)7-s + (0.309 + 0.951i)8-s + 3.45i·10-s + (3.21 − 0.797i)11-s + (−2.39 − 3.28i)13-s + (−0.951 + 0.309i)14-s + (−0.809 − 0.587i)16-s + (5.58 + 4.05i)17-s + (3.73 − 1.21i)19-s + (−2.03 − 2.79i)20-s + (−2.13 + 2.53i)22-s + 0.504i·23-s + ⋯
L(s)  = 1  + (−0.572 + 0.415i)2-s + (0.154 − 0.475i)4-s + (0.909 − 1.25i)5-s + (0.359 + 0.116i)7-s + (0.109 + 0.336i)8-s + 1.09i·10-s + (0.970 − 0.240i)11-s + (−0.662 − 0.912i)13-s + (−0.254 + 0.0825i)14-s + (−0.202 − 0.146i)16-s + (1.35 + 0.983i)17-s + (0.856 − 0.278i)19-s + (−0.454 − 0.625i)20-s + (−0.455 + 0.541i)22-s + 0.105i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.815 + 0.578i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.815 + 0.578i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1386\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 11\)
Sign: $0.815 + 0.578i$
Analytic conductor: \(11.0672\)
Root analytic conductor: \(3.32675\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1386} (701, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1386,\ (\ :1/2),\ 0.815 + 0.578i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.681125178\)
\(L(\frac12)\) \(\approx\) \(1.681125178\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.809 - 0.587i)T \)
3 \( 1 \)
7 \( 1 + (-0.951 - 0.309i)T \)
11 \( 1 + (-3.21 + 0.797i)T \)
good5 \( 1 + (-2.03 + 2.79i)T + (-1.54 - 4.75i)T^{2} \)
13 \( 1 + (2.39 + 3.28i)T + (-4.01 + 12.3i)T^{2} \)
17 \( 1 + (-5.58 - 4.05i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-3.73 + 1.21i)T + (15.3 - 11.1i)T^{2} \)
23 \( 1 - 0.504iT - 23T^{2} \)
29 \( 1 + (2.73 - 8.40i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (0.241 - 0.175i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-1.61 + 4.96i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-0.329 - 1.01i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 6.11iT - 43T^{2} \)
47 \( 1 + (-3.35 + 1.08i)T + (38.0 - 27.6i)T^{2} \)
53 \( 1 + (-1.04 - 1.43i)T + (-16.3 + 50.4i)T^{2} \)
59 \( 1 + (9.23 + 3.00i)T + (47.7 + 34.6i)T^{2} \)
61 \( 1 + (5.18 - 7.13i)T + (-18.8 - 58.0i)T^{2} \)
67 \( 1 + 7.92T + 67T^{2} \)
71 \( 1 + (-6.21 + 8.56i)T + (-21.9 - 67.5i)T^{2} \)
73 \( 1 + (-0.506 - 0.164i)T + (59.0 + 42.9i)T^{2} \)
79 \( 1 + (8.97 + 12.3i)T + (-24.4 + 75.1i)T^{2} \)
83 \( 1 + (-7.03 - 5.11i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + 13.3iT - 89T^{2} \)
97 \( 1 + (7.38 - 5.36i)T + (29.9 - 92.2i)T^{2} \)
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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

−9.218297640086239907031004825544, −8.918089022603916953544376726537, −7.950057254117608503550472043607, −7.26295116778504383721319488380, −5.94072854836925958606284045692, −5.54182871906542205410798436004, −4.74441951396706785746933523651, −3.35294373317557440485443793594, −1.75686921047795307626788815416, −0.975835706452721639311521348906, 1.36157848837457403094993268875, 2.40062514603023314613490595395, 3.26693077866660730107556648171, 4.42845553666595917394179526798, 5.67509812428063235431533083723, 6.55785302304723155957167004672, 7.27480637043534953636423406198, 7.907850132309824905969967894694, 9.329873362249110590098946553787, 9.647065612735199602374050971645

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