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Properties

Label 2-2268-63.38-c1-0-19
Degree $2$
Conductor $2268$
Sign $0.805 + 0.592i$
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  + (−0.385 + 0.667i)5-s + (2.37 − 1.17i)7-s + (−4.68 + 2.70i)11-s + (4.57 − 2.64i)13-s + (2.85 − 4.94i)17-s + (0.535 − 0.309i)19-s + (−5.83 − 3.37i)23-s + (2.20 + 3.81i)25-s + (6.99 + 4.03i)29-s − 0.580i·31-s + (−0.129 + 2.03i)35-s + (−4.53 − 7.86i)37-s + (−4.29 − 7.44i)41-s + (−1.70 + 2.94i)43-s + 0.770·47-s + ⋯
L(s)  = 1  + (−0.172 + 0.298i)5-s + (0.895 − 0.444i)7-s + (−1.41 + 0.814i)11-s + (1.26 − 0.732i)13-s + (0.692 − 1.19i)17-s + (0.122 − 0.0709i)19-s + (−1.21 − 0.702i)23-s + (0.440 + 0.763i)25-s + (1.29 + 0.749i)29-s − 0.104i·31-s + (−0.0218 + 0.344i)35-s + (−0.746 − 1.29i)37-s + (−0.670 − 1.16i)41-s + (−0.259 + 0.449i)43-s + 0.112·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.805 + 0.592i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.805 + 0.592i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2268\)    =    \(2^{2} \cdot 3^{4} \cdot 7\)
Sign: $0.805 + 0.592i$
Analytic conductor: \(18.1100\)
Root analytic conductor: \(4.25559\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2268} (269, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2268,\ (\ :1/2),\ 0.805 + 0.592i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.810666050\)
\(L(\frac12)\) \(\approx\) \(1.810666050\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.37 + 1.17i)T \)
good5 \( 1 + (0.385 - 0.667i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (4.68 - 2.70i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-4.57 + 2.64i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-2.85 + 4.94i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.535 + 0.309i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (5.83 + 3.37i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-6.99 - 4.03i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + 0.580iT - 31T^{2} \)
37 \( 1 + (4.53 + 7.86i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (4.29 + 7.44i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.70 - 2.94i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 0.770T + 47T^{2} \)
53 \( 1 + (-6.63 - 3.83i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 - 13.7T + 59T^{2} \)
61 \( 1 + 4.12iT - 61T^{2} \)
67 \( 1 - 13.5T + 67T^{2} \)
71 \( 1 + 2.25iT - 71T^{2} \)
73 \( 1 + (-1.96 - 1.13i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + 8.07T + 79T^{2} \)
83 \( 1 + (1.92 - 3.33i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (2.59 + 4.5i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-12.1 - 7.01i)T + (48.5 + 84.0i)T^{2} \)
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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

−8.705064468105809902413034963617, −8.143471650696307143434904204972, −7.43056814477633471950060931463, −6.86034334825294001838704812023, −5.50538138340386016268399903213, −5.13699230744062876311225127221, −4.07321230336455475201153217192, −3.11090167479444745975851528388, −2.09895984866289718289004328597, −0.73614468930319055385945416149, 1.12363661231682330653699654504, 2.19977714627142705693705814871, 3.37967119581666645822548364037, 4.26765675617753195802291901917, 5.23329730009115779964803244720, 5.85668427555210285440171392142, 6.64612868087001952165370469431, 8.085252063764047482695225968479, 8.194449803857640708416090879362, 8.712369697770468648307605046224

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