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Properties

Label 2-2268-252.167-c0-0-12
Degree $2$
Conductor $2268$
Sign $-0.0871 + 0.996i$
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  + (−0.258 − 0.965i)2-s + (−0.866 + 0.499i)4-s + (0.866 − 0.5i)7-s + (0.707 + 0.707i)8-s + (−0.258 − 0.448i)11-s + (−0.707 − 0.707i)14-s + (0.500 − 0.866i)16-s + (−0.366 + 0.366i)22-s + (0.707 − 1.22i)23-s + (0.5 + 0.866i)25-s + (−0.5 + 0.866i)28-s + (−1.22 + 0.707i)29-s + (−0.965 − 0.258i)32-s + 1.73·37-s + (0.866 − 0.5i)43-s + (0.448 + 0.258i)44-s + ⋯
L(s)  = 1  + (−0.258 − 0.965i)2-s + (−0.866 + 0.499i)4-s + (0.866 − 0.5i)7-s + (0.707 + 0.707i)8-s + (−0.258 − 0.448i)11-s + (−0.707 − 0.707i)14-s + (0.500 − 0.866i)16-s + (−0.366 + 0.366i)22-s + (0.707 − 1.22i)23-s + (0.5 + 0.866i)25-s + (−0.5 + 0.866i)28-s + (−1.22 + 0.707i)29-s + (−0.965 − 0.258i)32-s + 1.73·37-s + (0.866 − 0.5i)43-s + (0.448 + 0.258i)44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0871 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0871 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2268\)    =    \(2^{2} \cdot 3^{4} \cdot 7\)
Sign: $-0.0871 + 0.996i$
Analytic conductor: \(1.13187\)
Root analytic conductor: \(1.06389\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2268} (1511, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2268,\ (\ :0),\ -0.0871 + 0.996i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.034384052\)
\(L(\frac12)\) \(\approx\) \(1.034384052\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.258 + 0.965i)T \)
3 \( 1 \)
7 \( 1 + (-0.866 + 0.5i)T \)
good5 \( 1 + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.258 + 0.448i)T + (-0.5 + 0.866i)T^{2} \)
13 \( 1 + (0.5 + 0.866i)T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 + (-0.707 + 1.22i)T + (-0.5 - 0.866i)T^{2} \)
29 \( 1 + (1.22 - 0.707i)T + (0.5 - 0.866i)T^{2} \)
31 \( 1 + (-0.5 - 0.866i)T^{2} \)
37 \( 1 - 1.73T + T^{2} \)
41 \( 1 + (-0.5 - 0.866i)T^{2} \)
43 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
47 \( 1 + (0.5 - 0.866i)T^{2} \)
53 \( 1 + 1.93iT - T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \)
71 \( 1 - 1.93T + T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \)
83 \( 1 + (0.5 - 0.866i)T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + (0.5 - 0.866i)T^{2} \)
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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

−9.050716138574731087306870687636, −8.375309797721427020229224676898, −7.69248255533790781423757528139, −6.90493596845983963406939071769, −5.57054250554653271857398773354, −4.84571140127425227615399944438, −4.03588064868000120881986775735, −3.10619868684909420804412692441, −2.07380927584292872184408146141, −0.929682143265937135147022561357, 1.32470931576504892362420045390, 2.61033707195885073236433122931, 4.10224573903107779714782927856, 4.76889316671184652388922832133, 5.62258129401992267151865755718, 6.19195611297652923253820468572, 7.43948542258157157417132923610, 7.64065635388920878455822876195, 8.579897997055495645616367747640, 9.253076690162030301501300776450

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