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Properties

Label 2-1386-7.2-c1-0-9
Degree $2$
Conductor $1386$
Sign $0.749 - 0.661i$
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 + (−0.292 + 0.507i)5-s + (1.62 + 2.09i)7-s − 0.999·8-s + (0.292 + 0.507i)10-s + (0.5 + 0.866i)11-s − 3.82·13-s + (2.62 − 0.358i)14-s + (−0.5 + 0.866i)16-s + (1.82 + 3.16i)17-s + (0.292 − 0.507i)19-s + 0.585·20-s + 0.999·22-s + (−3.12 + 5.40i)23-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (−0.130 + 0.226i)5-s + (0.612 + 0.790i)7-s − 0.353·8-s + (0.0926 + 0.160i)10-s + (0.150 + 0.261i)11-s − 1.06·13-s + (0.700 − 0.0958i)14-s + (−0.125 + 0.216i)16-s + (0.443 + 0.768i)17-s + (0.0671 − 0.116i)19-s + 0.130·20-s + 0.213·22-s + (−0.650 + 1.12i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.749 - 0.661i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.749 - 0.661i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1386\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 11\)
Sign: $0.749 - 0.661i$
Analytic conductor: \(11.0672\)
Root analytic conductor: \(3.32675\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1386} (793, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1386,\ (\ :1/2),\ 0.749 - 0.661i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.652456848\)
\(L(\frac12)\) \(\approx\) \(1.652456848\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.62 - 2.09i)T \)
11 \( 1 + (-0.5 - 0.866i)T \)
good5 \( 1 + (0.292 - 0.507i)T + (-2.5 - 4.33i)T^{2} \)
13 \( 1 + 3.82T + 13T^{2} \)
17 \( 1 + (-1.82 - 3.16i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.292 + 0.507i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.12 - 5.40i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 2.65T + 29T^{2} \)
31 \( 1 + (-2 - 3.46i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-4.70 + 8.15i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 5.41T + 41T^{2} \)
43 \( 1 + 5.65T + 43T^{2} \)
47 \( 1 + (5.24 - 9.08i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-3.94 - 6.84i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (2.79 + 4.83i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (5.91 - 10.2i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.37 + 2.38i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 11.0T + 71T^{2} \)
73 \( 1 + (-4.70 - 8.15i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-6.62 + 11.4i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 12.1T + 83T^{2} \)
89 \( 1 + (-6.24 + 10.8i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 3.82T + 97T^{2} \)
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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

−9.594079084465275851841596276400, −9.159165559265680282000345380604, −7.992371786994175024909593942729, −7.38289653141203186243926056485, −6.13865218717778328462025436062, −5.38039568567185409384845540927, −4.58507277031259446959401272697, −3.52803436411245840570195939088, −2.49431033654741987803175642693, −1.52355265928822658220569050095, 0.60992088554666811325275681464, 2.35221743658253561645517991563, 3.61974060270454149527275859121, 4.63023384219698788269409144239, 5.06810655698789543555797692340, 6.30926385504824540853227487173, 7.01000941213697228368253459091, 7.941289106297893916730790295127, 8.287353469270580339794645803518, 9.518168000837738492076069693433

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