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Properties

Label 2-2268-84.83-c0-0-4
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

\(L(\frac{1}{2})\) \(\approx\) \(0.6776734196\)
\(L(\frac12)\) \(\approx\) \(0.6776734196\)
\(L(1)\) 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

−9.131145678906022624079289313854, −8.150540743898908832915529916411, −7.67487646132822671820001499438, −6.95325736985320271669209316228, −6.18029093126702965573749876514, −5.27895468684522322010884974587, −4.25377382597382936665630537466, −3.10523523773104528916494398548, −2.01096185849883498878181829577, −0.67428576824469390893144468412, 1.36745054032952608524540755017, 2.56571707932231761732010390153, 3.15192038663626936733336215520, 4.52332853197185583558295428209, 5.61331157669235624663230170093, 6.27349546219334578834723193992, 7.29766735729493335593525855958, 7.86163230470940582080745008716, 8.815727586821140923094503940490, 9.146365789043012499893729696002

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