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Properties

Label 2-2268-63.58-c1-0-16
Degree $2$
Conductor $2268$
Sign $0.966 + 0.257i$
Analytic cond. $18.1100$
Root an. cond. $4.25559$
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  + (−0.433 − 0.750i)5-s + (2.32 + 1.26i)7-s + (−1.75 + 3.04i)11-s + (0.933 − 1.61i)13-s + (−3.25 − 5.64i)17-s + (2.69 − 4.66i)19-s + (4.32 + 7.48i)23-s + (2.12 − 3.67i)25-s + (1.75 + 3.04i)29-s − 1.86·31-s + (−0.0576 − 2.29i)35-s + (1.39 − 2.40i)37-s + (5.19 − 8.99i)41-s + (2.89 + 5.00i)43-s − 6.16·47-s + ⋯
L(s)  = 1  + (−0.193 − 0.335i)5-s + (0.878 + 0.478i)7-s + (−0.529 + 0.917i)11-s + (0.258 − 0.448i)13-s + (−0.790 − 1.36i)17-s + (0.617 − 1.06i)19-s + (0.901 + 1.56i)23-s + (0.424 − 0.735i)25-s + (0.326 + 0.565i)29-s − 0.335·31-s + (−0.00975 − 0.387i)35-s + (0.228 − 0.395i)37-s + (0.810 − 1.40i)41-s + (0.440 + 0.763i)43-s − 0.898·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.966 + 0.257i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.966 + 0.257i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2268\)    =    \(2^{2} \cdot 3^{4} \cdot 7\)
Sign: $0.966 + 0.257i$
Analytic conductor: \(18.1100\)
Root analytic conductor: \(4.25559\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2268} (2053, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2268,\ (\ :1/2),\ 0.966 + 0.257i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.873369943\)
\(L(\frac12)\) \(\approx\) \(1.873369943\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 + (-2.32 - 1.26i)T \)
good5 \( 1 + (0.433 + 0.750i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (1.75 - 3.04i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.933 + 1.61i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (3.25 + 5.64i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.69 + 4.66i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-4.32 - 7.48i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.75 - 3.04i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 1.86T + 31T^{2} \)
37 \( 1 + (-1.39 + 2.40i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-5.19 + 8.99i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.89 - 5.00i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 6.16T + 47T^{2} \)
53 \( 1 + (-2.80 - 4.85i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 - 5.64T + 59T^{2} \)
61 \( 1 - 10.2T + 61T^{2} \)
67 \( 1 + 1.35T + 67T^{2} \)
71 \( 1 + 2.08T + 71T^{2} \)
73 \( 1 + (3.62 + 6.27i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 - 11.6T + 79T^{2} \)
83 \( 1 + (-3.43 - 5.94i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (3.28 - 5.68i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-1.64 - 2.85i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.127552284208444295135201584468, −8.209347023339057083957447865118, −7.39780720347860874131478618859, −6.94641966075365698916301018870, −5.52186851800915367010662683017, −5.04833044092869644913397322872, −4.38767467036004153083287504378, −3.01529848766558696072314774568, −2.19069090145038826054075113274, −0.847640151437315654159263708676, 0.990870437271577440937643639284, 2.18545866807943261625175178281, 3.35852857783684322860179886413, 4.17170707704862405338806710392, 5.02153655321951335146433287735, 5.99812865490621098455068792703, 6.70291680393608114658855911752, 7.61979050454779641018686169776, 8.360358156431733162164057298957, 8.735001561049484945770388068622

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