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Properties

Label 2-2268-63.34-c0-0-1
Degree $2$
Conductor $2268$
Sign $0.766 + 0.642i$
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  + (−1.5 + 0.866i)5-s + (0.5 − 0.866i)7-s − 1.73i·17-s + (1 − 1.73i)25-s + 1.73i·35-s + 37-s + (1.5 − 0.866i)41-s + (−0.5 + 0.866i)43-s + (1.5 + 0.866i)47-s + (−0.499 − 0.866i)49-s + (1.5 − 0.866i)59-s + (−1 − 1.73i)67-s + (0.5 − 0.866i)79-s + (−1.5 − 0.866i)83-s + (1.49 + 2.59i)85-s + ⋯
L(s)  = 1  + (−1.5 + 0.866i)5-s + (0.5 − 0.866i)7-s − 1.73i·17-s + (1 − 1.73i)25-s + 1.73i·35-s + 37-s + (1.5 − 0.866i)41-s + (−0.5 + 0.866i)43-s + (1.5 + 0.866i)47-s + (−0.499 − 0.866i)49-s + (1.5 − 0.866i)59-s + (−1 − 1.73i)67-s + (0.5 − 0.866i)79-s + (−1.5 − 0.866i)83-s + (1.49 + 2.59i)85-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2268\)    =    \(2^{2} \cdot 3^{4} \cdot 7\)
Sign: $0.766 + 0.642i$
Analytic conductor: \(1.13187\)
Root analytic conductor: \(1.06389\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2268} (1189, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2268,\ (\ :0),\ 0.766 + 0.642i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8766844331\)
\(L(\frac12)\) \(\approx\) \(0.8766844331\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−9.078949149296364721844578620518, −8.082530694953781923856075795120, −7.41449408425566771646105905193, −7.21506058530878837167490205310, −6.16629090741261463058074659633, −4.85775703499821089842068727015, −4.25458126245158779272629838271, −3.41615097084552867765963548166, −2.54753201292880288124664834330, −0.71508009158832481173920418565, 1.22125701575781913111418434407, 2.55028956964752775882841536748, 3.88614105807745036424933551367, 4.28584064779381969175476233229, 5.32008612071121046338432533705, 6.00753273252048260949815062299, 7.21036665033261675684970223857, 7.945544919764413663153684362676, 8.535340671225328278545687727690, 8.898364953839260081344464142834

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