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Properties

Label 2-1386-77.54-c1-0-18
Degree $2$
Conductor $1386$
Sign $0.998 - 0.0602i$
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  + (−0.866 − 0.5i)2-s + (0.499 + 0.866i)4-s + (0.606 + 0.350i)5-s + (1.82 + 1.91i)7-s − 0.999i·8-s + (−0.350 − 0.606i)10-s + (2.95 − 1.50i)11-s + 7.03·13-s + (−0.629 − 2.56i)14-s + (−0.5 + 0.866i)16-s + (−0.308 − 0.535i)17-s + (−0.391 + 0.678i)19-s + 0.700i·20-s + (−3.31 − 0.176i)22-s + (−2.63 + 4.56i)23-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (0.249 + 0.433i)4-s + (0.271 + 0.156i)5-s + (0.691 + 0.722i)7-s − 0.353i·8-s + (−0.110 − 0.191i)10-s + (0.891 − 0.453i)11-s + 1.95·13-s + (−0.168 − 0.686i)14-s + (−0.125 + 0.216i)16-s + (−0.0749 − 0.129i)17-s + (−0.0898 + 0.155i)19-s + 0.156i·20-s + (−0.706 − 0.0375i)22-s + (−0.549 + 0.951i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0602i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 - 0.0602i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1386\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 11\)
Sign: $0.998 - 0.0602i$
Analytic conductor: \(11.0672\)
Root analytic conductor: \(3.32675\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1386} (901, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1386,\ (\ :1/2),\ 0.998 - 0.0602i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.614510128\)
\(L(\frac12)\) \(\approx\) \(1.614510128\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 + 0.5i)T \)
3 \( 1 \)
7 \( 1 + (-1.82 - 1.91i)T \)
11 \( 1 + (-2.95 + 1.50i)T \)
good5 \( 1 + (-0.606 - 0.350i)T + (2.5 + 4.33i)T^{2} \)
13 \( 1 - 7.03T + 13T^{2} \)
17 \( 1 + (0.308 + 0.535i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.391 - 0.678i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.63 - 4.56i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 2.09iT - 29T^{2} \)
31 \( 1 + (5.35 - 3.08i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.99 + 3.45i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 5.85T + 41T^{2} \)
43 \( 1 + 3.62iT - 43T^{2} \)
47 \( 1 + (-2.03 - 1.17i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-3.95 - 6.84i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-3.25 + 1.88i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.18 - 7.24i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.07 + 1.85i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 4.48T + 71T^{2} \)
73 \( 1 + (2.64 + 4.58i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (3.75 + 2.16i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 4.03T + 83T^{2} \)
89 \( 1 + (14.0 + 8.10i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 10.4iT - 97T^{2} \)
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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

−9.367768790363276876762458799349, −8.846718734252054446747316248643, −8.241616145623399507595647428746, −7.32484116478303469882260881324, −6.07736404190208682729558232352, −5.81199177339228995331731130310, −4.22702143980681432685578083136, −3.43554920947416314858000617107, −2.12175163306895071228025114145, −1.19605791111020472129306311865, 1.04616844337385700342666268718, 1.89899150643513216071324094718, 3.67138514935046027379266558151, 4.41715955537247336075572601483, 5.63442105355035231663704317789, 6.39038232380720730610763064601, 7.15602004152497516857703517404, 8.092008234574080095148610054638, 8.694561821107625626163042380777, 9.447726755425184087146821563733

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