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Properties

Label 2-2268-7.4-c1-0-15
Degree $2$
Conductor $2268$
Sign $0.844 - 0.535i$
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 + (1.55 + 2.14i)7-s + (−0.792 + 1.37i)11-s + 5.04·13-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 + 8.89·29-s + (0.575 − 0.996i)31-s + (−1.10 + 2.49i)35-s + (4.07 + 7.06i)37-s − 7.74·41-s − 2.53·43-s + (4.24 + 7.35i)47-s + ⋯
L(s)  = 1  + (0.230 + 0.399i)5-s + (0.588 + 0.808i)7-s + (−0.239 + 0.414i)11-s + 1.40·13-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 + 1.65·29-s + (0.103 − 0.179i)31-s + (−0.187 + 0.421i)35-s + (0.670 + 1.16i)37-s − 1.20·41-s − 0.386·43-s + (0.619 + 1.07i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.844 - 0.535i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.844 - 0.535i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2268\)    =    \(2^{2} \cdot 3^{4} \cdot 7\)
Sign: $0.844 - 0.535i$
Analytic conductor: \(18.1100\)
Root analytic conductor: \(4.25559\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2268} (1621, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2268,\ (\ :1/2),\ 0.844 - 0.535i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.201906425\)
\(L(\frac12)\) \(\approx\) \(2.201906425\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.55 - 2.14i)T \)
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 \( 1 - 5.04T + 13T^{2} \)
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 \( 1 - 8.89T + 29T^{2} \)
31 \( 1 + (-0.575 + 0.996i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-4.07 - 7.06i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 7.74T + 41T^{2} \)
43 \( 1 + 2.53T + 43T^{2} \)
47 \( 1 + (-4.24 - 7.35i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (2.41 - 4.17i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-1.93 + 3.35i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.82 - 8.35i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.837 + 1.45i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 14.2T + 71T^{2} \)
73 \( 1 + (-3.04 + 5.27i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-4.15 - 7.19i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 14.2T + 83T^{2} \)
89 \( 1 + (6.69 + 11.5i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 5.34T + 97T^{2} \)
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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

−8.909788962153003972726898695788, −8.399950601008314682388452319534, −7.68274298860313635193884333845, −6.53550072486982352839339765284, −6.13688902037662803828947624806, −5.05597508528773668418643829364, −4.42086482656751609872853354313, −3.07680635512588918684338007663, −2.40526564907108299572570650382, −1.11564810925173207104951502788, 0.980027760943562536926010481942, 1.78238360323112623639206820727, 3.40113008062538139626128033886, 3.94940872543825876171662985051, 5.03891966644152950069233382226, 5.77877928911327608770450024795, 6.55133948552479957219784378293, 7.54372663728064898948359059329, 8.329683166252026508876717506646, 8.670483383023096172463384464660

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