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Properties

Label 2-2214-369.286-c1-0-17
Degree $2$
Conductor $2214$
Sign $0.933 - 0.359i$
Analytic cond. $17.6788$
Root an. cond. $4.20462$
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 + (1.58 + 2.74i)5-s + (1.86 + 1.07i)7-s − 0.999·8-s + 3.17·10-s + (0.206 + 0.119i)11-s + (6.02 − 3.47i)13-s + (1.86 − 1.07i)14-s + (−0.5 + 0.866i)16-s + 4.96i·17-s + 6.10i·19-s + (1.58 − 2.74i)20-s + (0.206 − 0.119i)22-s + (1.71 + 2.97i)23-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (0.709 + 1.22i)5-s + (0.704 + 0.406i)7-s − 0.353·8-s + 1.00·10-s + (0.0623 + 0.0360i)11-s + (1.67 − 0.964i)13-s + (0.497 − 0.287i)14-s + (−0.125 + 0.216i)16-s + 1.20i·17-s + 1.40i·19-s + (0.354 − 0.614i)20-s + (0.0440 − 0.0254i)22-s + (0.358 + 0.620i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2214 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.933 - 0.359i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2214 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.933 - 0.359i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2214\)    =    \(2 \cdot 3^{3} \cdot 41\)
Sign: $0.933 - 0.359i$
Analytic conductor: \(17.6788\)
Root analytic conductor: \(4.20462\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2214} (1639, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2214,\ (\ :1/2),\ 0.933 - 0.359i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.719371320\)
\(L(\frac12)\) \(\approx\) \(2.719371320\)
\(L(\frac{3}{2})\) 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 \( 1 \)
41 \( 1 + (-4.10 - 4.91i)T \)
good5 \( 1 + (-1.58 - 2.74i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (-1.86 - 1.07i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.206 - 0.119i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-6.02 + 3.47i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 - 4.96iT - 17T^{2} \)
19 \( 1 - 6.10iT - 19T^{2} \)
23 \( 1 + (-1.71 - 2.97i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.10 + 1.79i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (4.95 + 8.58i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 10.7T + 37T^{2} \)
43 \( 1 + (1.63 - 2.82i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-5.37 - 3.10i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 - 1.18iT - 53T^{2} \)
59 \( 1 + (5.69 + 9.85i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.854 + 1.48i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4.46 + 2.57i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 8.31iT - 71T^{2} \)
73 \( 1 - 12.0T + 73T^{2} \)
79 \( 1 + (-0.375 - 0.216i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-1.97 + 3.42i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 4.54iT - 89T^{2} \)
97 \( 1 + (-6.84 - 3.95i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.252834818900711344800851854751, −8.278308073561860766774399843794, −7.70958557428238502099993790123, −6.36203487390447063594877504432, −5.94845536978599381017474296836, −5.31293877644011890373732630050, −3.79659937825258436170462916156, −3.43439687863290717864002197363, −2.16876200036391851996308151727, −1.48488300164501479456572348933, 0.910188338238698107211318349967, 1.92291889084777466942284476825, 3.43860491092405475859819140071, 4.46442535928035745151069629143, 5.03454374013455727188389183454, 5.66066144852592113360319529762, 6.76651799175838073770396776154, 7.22667987289946260555144813764, 8.471055427716857429471817653164, 8.985448645628706054431440876229

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