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Properties

Label 2-2268-84.83-c0-0-3
Degree $2$
Conductor $2268$
Sign $0.5 - 0.866i$
Analytic cond. $1.13187$
Root an. cond. $1.06389$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.965 + 0.258i)2-s + (0.866 + 0.499i)4-s + i·7-s + (0.707 + 0.707i)8-s + 0.517·11-s + (−0.258 + 0.965i)14-s + (0.500 + 0.866i)16-s + (0.499 + 0.133i)22-s − 1.41·23-s − 25-s + (−0.499 + 0.866i)28-s − 1.41i·29-s + (0.258 + 0.965i)32-s + 1.73·37-s + i·43-s + (0.448 + 0.258i)44-s + ⋯
L(s)  = 1  + (0.965 + 0.258i)2-s + (0.866 + 0.499i)4-s + i·7-s + (0.707 + 0.707i)8-s + 0.517·11-s + (−0.258 + 0.965i)14-s + (0.500 + 0.866i)16-s + (0.499 + 0.133i)22-s − 1.41·23-s − 25-s + (−0.499 + 0.866i)28-s − 1.41i·29-s + (0.258 + 0.965i)32-s + 1.73·37-s + i·43-s + (0.448 + 0.258i)44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.5 - 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.5 - 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2268\)    =    \(2^{2} \cdot 3^{4} \cdot 7\)
Sign: $0.5 - 0.866i$
Analytic conductor: \(1.13187\)
Root analytic conductor: \(1.06389\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2268} (2267, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2268,\ (\ :0),\ 0.5 - 0.866i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.202527959\)
\(L(\frac12)\) \(\approx\) \(2.202527959\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.965 - 0.258i)T \)
3 \( 1 \)
7 \( 1 - iT \)
good5 \( 1 + T^{2} \)
11 \( 1 - 0.517T + T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 + 1.41T + T^{2} \)
29 \( 1 + 1.41iT - T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 - 1.73T + T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 - iT - T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + 1.93iT - T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 - 1.73iT - T^{2} \)
71 \( 1 - 1.93T + T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + 1.73iT - T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 - T^{2} \)
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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

−9.420427741004923560304987771480, −8.161062751883533075639947500273, −7.963956295826578732604519444786, −6.72215594402366733270807540438, −6.05973730277574925258684393373, −5.55665097906648102250316387920, −4.46578304749411391955400064187, −3.80721743271312011302035514214, −2.66616893357478130048225729603, −1.89348397835900178839870740312, 1.23703269182850006572824317800, 2.35802628459458033757307753516, 3.61539445011157566808830203093, 4.06713215777185324208041315275, 4.96478765507169149477693825858, 5.95076490069065182648390937274, 6.58063346369173750639182702134, 7.42747662184814106688470651328, 8.037068799975528522600317836297, 9.327521980001951466495678344981

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