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Properties

Label 2-1386-7.4-c1-0-15
Degree $2$
Conductor $1386$
Sign $0.386 - 0.922i$
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 + (2.5 + 0.866i)7-s − 0.999·8-s + (−0.5 + 0.866i)11-s + 5·13-s + (0.500 + 2.59i)14-s + (−0.5 − 0.866i)16-s + (3 − 5.19i)17-s + (−1 − 1.73i)19-s − 0.999·22-s + (3 + 5.19i)23-s + (2.5 − 4.33i)25-s + (2.5 + 4.33i)26-s + (−1.99 + 1.73i)28-s − 3·29-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + (0.944 + 0.327i)7-s − 0.353·8-s + (−0.150 + 0.261i)11-s + 1.38·13-s + (0.133 + 0.694i)14-s + (−0.125 − 0.216i)16-s + (0.727 − 1.26i)17-s + (−0.229 − 0.397i)19-s − 0.213·22-s + (0.625 + 1.08i)23-s + (0.5 − 0.866i)25-s + (0.490 + 0.849i)26-s + (−0.377 + 0.327i)28-s − 0.557·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1386\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 11\)
Sign: $0.386 - 0.922i$
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: \((2,\ 1386,\ (\ :1/2),\ 0.386 - 0.922i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.324242512\)
\(L(\frac12)\) \(\approx\) \(2.324242512\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.5 - 0.866i)T \)
11 \( 1 + (0.5 - 0.866i)T \)
good5 \( 1 + (-2.5 + 4.33i)T^{2} \)
13 \( 1 - 5T + 13T^{2} \)
17 \( 1 + (-3 + 5.19i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1 + 1.73i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3 - 5.19i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 3T + 29T^{2} \)
31 \( 1 + (4 - 6.92i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (1 + 1.73i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 6T + 41T^{2} \)
43 \( 1 + 4T + 43T^{2} \)
47 \( 1 + (-3 - 5.19i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (6 - 10.3i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.5 - 2.59i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.5 - 6.06i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-6.5 + 11.2i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 12T + 71T^{2} \)
73 \( 1 + (-5 + 8.66i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.5 - 0.866i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 6T + 83T^{2} \)
89 \( 1 + (-3 - 5.19i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 13T + 97T^{2} \)
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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

−9.379555336746172130787091316693, −8.889022219218410515695836530303, −7.961537165579687276118566942626, −7.36251084211016189474548588857, −6.41716003807162379629165287649, −5.44033758188113459763450083063, −4.91270848811148986621272277109, −3.82410473994110530795284801153, −2.76390606311101058541979669854, −1.27505483400045629054360020248, 1.06698112310847881631634707623, 2.04283183714581064365589456438, 3.50870654016409637107085711208, 4.05808537982217627440062105224, 5.21335681472076505683043377906, 5.89666689105492550792566417797, 6.89413141030597965906877147854, 8.147951823831937719370681748093, 8.451100725055314291334004970982, 9.542029117643362321591924420731

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