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Properties

Label 2-2268-63.5-c1-0-19
Degree $2$
Conductor $2268$
Sign $0.110 + 0.993i$
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  + (−1.07 − 1.87i)5-s + (−2.20 + 1.46i)7-s + (1.15 + 0.667i)11-s + (−2.03 − 1.17i)13-s + (3.60 + 6.23i)17-s + (1.03 + 0.599i)19-s + (−2.08 + 1.20i)23-s + (0.167 − 0.290i)25-s + (5.32 − 3.07i)29-s − 8.21i·31-s + (5.11 + 2.53i)35-s + (2.57 − 4.45i)37-s + (2.23 − 3.87i)41-s + (0.332 + 0.575i)43-s + 2.15·47-s + ⋯
L(s)  = 1  + (−0.482 − 0.836i)5-s + (−0.832 + 0.553i)7-s + (0.348 + 0.201i)11-s + (−0.564 − 0.325i)13-s + (0.873 + 1.51i)17-s + (0.238 + 0.137i)19-s + (−0.434 + 0.250i)23-s + (0.0335 − 0.0580i)25-s + (0.988 − 0.570i)29-s − 1.47i·31-s + (0.865 + 0.429i)35-s + (0.423 − 0.732i)37-s + (0.349 − 0.604i)41-s + (0.0506 + 0.0878i)43-s + 0.315·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.110 + 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.110 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2268\)    =    \(2^{2} \cdot 3^{4} \cdot 7\)
Sign: $0.110 + 0.993i$
Analytic conductor: \(18.1100\)
Root analytic conductor: \(4.25559\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2268} (1349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2268,\ (\ :1/2),\ 0.110 + 0.993i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.093219469\)
\(L(\frac12)\) \(\approx\) \(1.093219469\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.20 - 1.46i)T \)
good5 \( 1 + (1.07 + 1.87i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.15 - 0.667i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.03 + 1.17i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-3.60 - 6.23i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.03 - 0.599i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.08 - 1.20i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-5.32 + 3.07i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + 8.21iT - 31T^{2} \)
37 \( 1 + (-2.57 + 4.45i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-2.23 + 3.87i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.332 - 0.575i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 2.15T + 47T^{2} \)
53 \( 1 + (11.0 - 6.37i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + 9.66T + 59T^{2} \)
61 \( 1 + 12.2iT - 61T^{2} \)
67 \( 1 + 3.73T + 67T^{2} \)
71 \( 1 + 11.4iT - 71T^{2} \)
73 \( 1 + (-9.07 + 5.23i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + 9.07T + 79T^{2} \)
83 \( 1 + (5.39 + 9.35i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-2.59 + 4.5i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (1.07 - 0.618i)T + (48.5 - 84.0i)T^{2} \)
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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

−8.866830229304496640036781525567, −8.016209835778304381829927024095, −7.56661156489269484592930585289, −6.20915265708549539634164959593, −5.90356918703161439525949478951, −4.75721502255512825030873521850, −3.99169975591978552521292590061, −3.08322193026071188202413241450, −1.86678963557726490159718749865, −0.44673875459435172138756392839, 1.05669949422575879739781534770, 2.88050157698597723421336674366, 3.17782200277772046161323892666, 4.30578869027092117266650996653, 5.18094225612268759951235990889, 6.33446544488558972360389805368, 6.99564529563492274602063110795, 7.38597772356890252353603153997, 8.374229293239008143545268258460, 9.394312487061982703241942704275

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