Properties

Label 2-2268-63.58-c1-0-23
Degree $2$
Conductor $2268$
Sign $-0.792 + 0.610i$
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  + (0.515 + 0.892i)5-s + (−2.63 − 0.277i)7-s + (−0.792 + 1.37i)11-s + (−2.52 + 4.37i)13-s + (2.58 + 4.47i)17-s + (−0.392 + 0.680i)19-s + (−2.93 − 5.07i)23-s + (1.96 − 3.40i)25-s + (−4.44 − 7.69i)29-s − 1.15·31-s + (−1.10 − 2.49i)35-s + (4.07 − 7.06i)37-s + (3.87 − 6.70i)41-s + (1.26 + 2.19i)43-s − 8.49·47-s + ⋯
L(s)  = 1  + (0.230 + 0.399i)5-s + (−0.994 − 0.104i)7-s + (−0.239 + 0.414i)11-s + (−0.700 + 1.21i)13-s + (0.626 + 1.08i)17-s + (−0.0901 + 0.156i)19-s + (−0.611 − 1.05i)23-s + (0.393 − 0.681i)25-s + (−0.825 − 1.42i)29-s − 0.206·31-s + (−0.187 − 0.421i)35-s + (0.670 − 1.16i)37-s + (0.604 − 1.04i)41-s + (0.193 + 0.334i)43-s − 1.23·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.792 + 0.610i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.792 + 0.610i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2268\)    =    \(2^{2} \cdot 3^{4} \cdot 7\)
Sign: $-0.792 + 0.610i$
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: \((2,\ 2268,\ (\ :1/2),\ -0.792 + 0.610i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1473387871\)
\(L(\frac12)\) \(\approx\) \(0.1473387871\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.63 + 0.277i)T \)
good5 \( 1 + (-0.515 - 0.892i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (0.792 - 1.37i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (2.52 - 4.37i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-2.58 - 4.47i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.392 - 0.680i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.93 + 5.07i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (4.44 + 7.69i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 1.15T + 31T^{2} \)
37 \( 1 + (-4.07 + 7.06i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-3.87 + 6.70i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.26 - 2.19i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 8.49T + 47T^{2} \)
53 \( 1 + (2.41 + 4.17i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + 3.87T + 59T^{2} \)
61 \( 1 + 9.64T + 61T^{2} \)
67 \( 1 + 1.67T + 67T^{2} \)
71 \( 1 + 14.2T + 71T^{2} \)
73 \( 1 + (-3.04 - 5.27i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + 8.31T + 79T^{2} \)
83 \( 1 + (7.12 + 12.3i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (6.69 - 11.5i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (2.67 + 4.63i)T + (-48.5 + 84.0i)T^{2} \)
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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

−8.833989541564215704389736332957, −7.82745632190863493456653975675, −7.14170229762901860918088043253, −6.28655622687429090991003424316, −5.87156937713134594763002200960, −4.48828391369540991442294000308, −3.90543016353479769247948618995, −2.70819707029224567578170324694, −1.92416816396848114354432765364, −0.04977216348824032139182074597, 1.32397848168406133457371702477, 2.96969558681581936274793006986, 3.21898709834007567057281650211, 4.69449606792489842569076367628, 5.47250655050357585632776631890, 6.01015045027182756075733918826, 7.17552973822063951500939687256, 7.66101804047046659180533184542, 8.636997523023183283157538700205, 9.535121869320172238800928907471

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