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Properties

Label 2-1386-77.10-c1-0-2
Degree $2$
Conductor $1386$
Sign $-0.997 + 0.0766i$
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 + (−1.09 + 0.629i)5-s + (2.11 + 1.59i)7-s + 0.999i·8-s + (0.629 − 1.09i)10-s + (−1.45 + 2.98i)11-s − 4.08·13-s + (−2.62 − 0.319i)14-s + (−0.5 − 0.866i)16-s + (1.60 − 2.77i)17-s + (3.81 + 6.60i)19-s + 1.25i·20-s + (−0.235 − 3.30i)22-s + (−4.12 − 7.14i)23-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (0.249 − 0.433i)4-s + (−0.487 + 0.281i)5-s + (0.799 + 0.600i)7-s + 0.353i·8-s + (0.199 − 0.344i)10-s + (−0.437 + 0.899i)11-s − 1.13·13-s + (−0.701 − 0.0854i)14-s + (−0.125 − 0.216i)16-s + (0.388 − 0.672i)17-s + (0.875 + 1.51i)19-s + 0.281i·20-s + (−0.0502 − 0.705i)22-s + (−0.860 − 1.49i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 + 0.0766i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.997 + 0.0766i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1386\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 11\)
Sign: $-0.997 + 0.0766i$
Analytic conductor: \(11.0672\)
Root analytic conductor: \(3.32675\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1386} (703, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1386,\ (\ :1/2),\ -0.997 + 0.0766i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4347669366\)
\(L(\frac12)\) \(\approx\) \(0.4347669366\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.11 - 1.59i)T \)
11 \( 1 + (1.45 - 2.98i)T \)
good5 \( 1 + (1.09 - 0.629i)T + (2.5 - 4.33i)T^{2} \)
13 \( 1 + 4.08T + 13T^{2} \)
17 \( 1 + (-1.60 + 2.77i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.81 - 6.60i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (4.12 + 7.14i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 3.54iT - 29T^{2} \)
31 \( 1 + (7.95 + 4.59i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.154 - 0.266i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 6.05T + 41T^{2} \)
43 \( 1 + 7.57iT - 43T^{2} \)
47 \( 1 + (4.07 - 2.35i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (2.39 - 4.15i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (2.36 + 1.36i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.755 + 1.30i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.69 - 2.92i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 3.50T + 71T^{2} \)
73 \( 1 + (-0.483 + 0.837i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (13.5 - 7.80i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 - 1.32T + 83T^{2} \)
89 \( 1 + (9.22 - 5.32i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 10.6iT - 97T^{2} \)
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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

−9.856237030242829878254630009644, −9.248112498528937508399786111093, −8.145870567255083249140564273855, −7.61931754876390224798897055877, −7.12262871029171857860200054863, −5.78385832961148256545518657250, −5.16055421302930319051954861688, −4.13951516434218115309182359407, −2.69478422246731129028378557852, −1.73322519822277411970392801034, 0.21391591246184046653383275121, 1.53654168789509208573733717406, 2.86128086622444976241157251067, 3.88796125832667187454006097254, 4.85982276533817976085387232908, 5.75288512749189446919996147833, 7.12847986986973879976883268921, 7.72642380509413811862715587670, 8.225425802133091319477101655875, 9.227596610080932155180688876827

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