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Properties

Label 2-2268-63.4-c1-0-21
Degree $2$
Conductor $2268$
Sign $-0.160 + 0.987i$
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  − 2.42·5-s + (0.710 − 2.54i)7-s + 4.70·11-s + (1.71 − 2.96i)13-s + (−0.851 + 1.47i)17-s + (−0.641 − 1.11i)19-s − 1.12·23-s + 0.861·25-s + (2.35 + 4.07i)29-s + (1.71 + 2.96i)31-s + (−1.72 + 6.17i)35-s + (−4.27 − 7.40i)37-s + (−1.85 + 3.21i)41-s + (−2.77 − 4.80i)43-s + (5.91 − 10.2i)47-s + ⋯
L(s)  = 1  − 1.08·5-s + (0.268 − 0.963i)7-s + 1.41·11-s + (0.474 − 0.821i)13-s + (−0.206 + 0.357i)17-s + (−0.147 − 0.254i)19-s − 0.234·23-s + 0.172·25-s + (0.436 + 0.756i)29-s + (0.307 + 0.532i)31-s + (−0.290 + 1.04i)35-s + (−0.702 − 1.21i)37-s + (−0.290 + 0.502i)41-s + (−0.422 − 0.732i)43-s + (0.862 − 1.49i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.160 + 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.160 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2268\)    =    \(2^{2} \cdot 3^{4} \cdot 7\)
Sign: $-0.160 + 0.987i$
Analytic conductor: \(18.1100\)
Root analytic conductor: \(4.25559\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2268} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2268,\ (\ :1/2),\ -0.160 + 0.987i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.282526890\)
\(L(\frac12)\) \(\approx\) \(1.282526890\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.710 + 2.54i)T \)
good5 \( 1 + 2.42T + 5T^{2} \)
11 \( 1 - 4.70T + 11T^{2} \)
13 \( 1 + (-1.71 + 2.96i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (0.851 - 1.47i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.641 + 1.11i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 1.12T + 23T^{2} \)
29 \( 1 + (-2.35 - 4.07i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1.71 - 2.96i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (4.27 + 7.40i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.85 - 3.21i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.77 + 4.80i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-5.91 + 10.2i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (5.13 - 8.88i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (2.06 + 3.57i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.62 + 8.01i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.56 - 9.63i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 14.9T + 71T^{2} \)
73 \( 1 + (1.06 - 1.85i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-7.26 + 12.5i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (4.21 + 7.29i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (8.04 + 13.9i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (8.12 + 14.0i)T + (-48.5 + 84.0i)T^{2} \)
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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

−8.553225510643408522239307035609, −8.177364396700463287074242777380, −7.13694641303221817303253330106, −6.78513346350565415153030137031, −5.65361422023884493500247211168, −4.55184989250896081922894726263, −3.86880750355639836279290855191, −3.33004189500800532882821924079, −1.62857791726058252989425496255, −0.49663260210324974916237500422, 1.28538984972666784261469635172, 2.47309816128636559499470223219, 3.71599524623527955100136116559, 4.23170354092425264510658564698, 5.19381050548921067903915438125, 6.36796808955725465314436787141, 6.70753046524975870186957177666, 7.962331931990610457537781767105, 8.336439648192951805330397673560, 9.227204013224578994705473279857

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